Intermediate mathematics/Numbers

= Numbers =

Scalars

 * See also:, ,

The basis of all of mathematics is the function. See.


 * Next(0)=1
 * Next(1)=2
 * Next(2)=3
 * Next(3)=4
 * Next(4)=5

We might express this by saying that One differs from nothing as two differs from one. This defines the (denoted $$\mathbb{N}_0$$). Natural numbers are those used for counting.


 * These have the convenient property of being . That means that if a<b and b<c then it follows that a<c. In fact they are . See.

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Integers
(See ) is defined as repeatedly calling the Next function, and its inverse is. But this leads to the ability to write equations like $$1-3=x$$ for which there is no answer among natural numbers. To provide an answer mathematicians generalize to the set of all (denoted $$\mathbb{Z}$$ because zahlen means count in german) which includes negative integers.


 * The is zero because x + 0 = x.


 * The absolute value or modulus of $x$ is defined as $$|x| = \left\{

\begin{array}{rl} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0. \end{array}\right. $$


 * (denoted $$\mathcal O_\mathbb{Q}$$) over the field of rational numbers. is defined below.


 * $Z_{n}$ or $$\mathbb{Z}/n\mathbb{Z}$$ is used to denote the set of.


 * is essentially arithmetic in the Z/nZ (which has n elements).


 * Consider the ring of integers Z and the ideal of even numbers, denoted by 2Z. Then the quotient ring Z / 2Z has only two elements, zero for the even numbers and one for the odd numbers; applying the definition, [z] = z + 2Z := {z + 2y: 2y ∈ 2Z}, where 2Z is the ideal of even numbers. It is naturally isomorphic to the finite field with two elements, F2. Intuitively: if you think of all the even numbers as 0, then every integer is either 0 (if it is even) or 1 (if it is odd and therefore differs from an even number by 1).


 * An is a special subset of a ring. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3.


 * A is an ideal $$I$$ in a ring $$R$$ that is generated by a single element $$a$$ of $$R$$ through multiplication by every element of $$R$$.


 * A is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number.


 * The study of integers is called.


 * $$a \mid b$$ means a divides b.


 * $$a \nmid b$$ means a does not divide b.


 * $$p^a \mid\mid n$$ means pa exactly divides n (i.e. pa divides n but pa+1 does not).


 * A prime number is a number that can only be divided by itself and one.


 * If a, b, c, and d are primes and x=abc and y=c2d then:


 * xy = *  = abc2d * c


 * Two integers a and b are said to be relatively prime, mutually prime, or coprime if the only positive integer that divides both of them is 1. Any prime number that divides one does not divide the other. This is equivalent to their greatest common divisor (gcd) being 1.


 * (See )

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Rational numbers
(See ) is defined as repeated addition, and its inverse is. But this leads to equations like $$3/2=x$$ for which there is no answer. The solution is to generalize to the set of (denoted $$\mathbb{Q}$$) which include fractions (See ). Any number which isnt rational is. See also


 * The set of all rational numbers except zero forms a which is a set of invertible elements.


 * Rational numbers form a because every non-zero element has an inverse. The ability to find the inverse of every element turns out to be quite useful. A great deal of time and effort has been spent trying to find division algebras.


 * Rational numbers form a . A field is a set on which addition, subtraction, multiplication, and division are defined. See below.


 * The is one because x * 1 = x.


 * . 1/0 exists nowhere on the . It does, however, exist on the (often called the extended complex plane) where it is surprisingly well behaved. See also  and.


 * $$\frac{1}{nothing}=everything$$


 * (Addition and multiplication are fast but division is slow .)

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Irrational and complex numbers
(See ) is defined as repeated multiplication, and its inverses are and. But this leads to multiple equations with no solutions:


 * Equations like $$\sqrt{2}=x.$$ The solution is to generalize to the set of (denoted $$\mathbb{A}$$). (See also  and algebraically closed.) To see a proof that the square root of two is irrational see.


 * Equations like $$2^{\sqrt{2}}=x$$ The solution (because x is ) is to generalize to the set of (denoted $$\mathbb{R}$$).




 * Equations like $$\sqrt{-1}=x$$ and $$e^x=-1.$$ The solution is to generalize to the set of (denoted $$\mathbb{C}$$) by defining i = sqrt(-1). A single complex number $$z=a+bi$$ consists of a real part a and an imaginary part bi (See ).  (denoted $$\mathbb{I}$$) often occur in equations involving change with respect to time. If friction is resistance to motion then imaginary friction would be resistance to change of motion wrt time. (In other words, imaginary friction would be mass.) In fact, in the equation for the  (given below),.


 * The of the complex number $$z=a+bi$$ is $$\overline{z}=a-bi.$$ (Not to be confused with the  of a vector.)


 * $$\frac{1}{a+bi} = \frac{1}{a+bi} \cdot \frac{a-bi}{a-bi} = \frac{a-bi}{a^2+b^2}$$


 * Complex numbers form an (K-algebra) because complex multiplication is.


 * $$\sqrt{-100} * \sqrt{-100} = 10i * 10i = -100 \neq \sqrt{-100 * -100}$$


 * The complex numbers are not . However the or  of a complex number is:


 * $$|z| = |a + ib| = \sqrt{a^2+b^2}$$


 * A Gaussian integer $n$ is a Gaussian prime if and only if either:
 * one of $a + bi$ is zero and absolute value of the other is a prime number of the form $a, b$ (with $n$ a nonnegative integer), or
 * both are nonzero and $4n + 3$ is a prime number (which will not be of the form $a^{2} + b^{2}$).


 * There are n solutions of $$\sqrt[n]{z}$$


 * 0^0 = 1. See.


 * $$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$$

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Hypercomplex numbers
Complex numbers can be used to represent and but only in 2 dimensions. like (denoted $$\mathbb{H}$$),  (denoted $$\mathbb{O}$$), and  (denoted $$\mathbb{S}$$) are one way to generalize complex numbers to some (but not all) higher dimensions.

A quaternion can be thought of as a complex number whose coefficients are themselves complex numbers (hence a hypercomplex number).


 * $$(a + b\boldsymbol{\hat{\imath}}) + (c + d\boldsymbol{\hat{\imath}})\boldsymbol{\hat{\jmath}} = a + b\boldsymbol{\hat{\imath}} + c\boldsymbol{\hat{\jmath}} + d\boldsymbol{\hat{\imath}\hat{\jmath}} =  a + b\boldsymbol{\hat{\imath}} + c\boldsymbol{\hat{\jmath}} + d\boldsymbol{\hat{k}}$$

Where


 * $$\boldsymbol{\hat{\imath}}^2 = \boldsymbol{\hat{\jmath}}^2 = \boldsymbol{\hat{k}}^2 = \boldsymbol{\hat{\imath}} \boldsymbol{\hat{\jmath}} \boldsymbol{\hat{k}} = -1$$

and
 * $$\begin{alignat}{2}

\boldsymbol{\hat{\imath}}\boldsymbol{\hat{\jmath}} & = \boldsymbol{\hat{k}}, & \qquad \boldsymbol{\hat{\jmath}}\boldsymbol{\hat{\imath}} & = -\boldsymbol{\hat{k}}, \\ \boldsymbol{\hat{\jmath}}\boldsymbol{\hat{k}} & = \boldsymbol{\hat{\imath}}, & \boldsymbol{\hat{k}}\boldsymbol{\hat{\jmath}} & = -\boldsymbol{\hat{\imath}}, \\ \boldsymbol{\hat{k}}\boldsymbol{\hat{\imath}} & = \boldsymbol{\hat{\jmath}}, & \boldsymbol{\hat{\imath}}\boldsymbol{\hat{k}} & = -\boldsymbol{\hat{\jmath}}. \end{alignat}$$

Any real finite-dimensional over the reals must be:
 * isomorphic to R or C if and commutative (equivalently: associative and commutative)
 * isomorphic to the quaternions if noncommutative but associative
 * isomorphic to the octonions if non-associative but alternative.

The following is known about the dimension of a finite-dimensional division algebra A over a field K:
 * dim A = 1 if K is algebraically closed,
 * dim A = 1, 2, 4 or 8 if K is, and
 * If K is neither algebraically nor real closed, then there are infinitely many dimensions in which there exist division algebras over K.

(hyperbolic complex numbers) are similar to complex numbers except that i2 = +1.

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Tetration
is defined as repeated exponentiation and its inverses are called super-root and super-logarithm.



\begin{matrix}

{}^{b}a &

= &

\underbrace{a^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} &

= &

a\uparrow\uparrow b

= &

\underbrace{a\uparrow (a\uparrow(\dots\uparrow a))} &

\\

& & b\mbox{ copies of }a

&

& & b\mbox{ copies of }a

\end{matrix}

$$

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Hyperreal numbers

 * See also:

When a quantity, like the charge of a single electron, becomes so small that it is insignificant we, quite justifiably, treat it as though it were zero. A quantity that can be treated as though it were zero, even though it very definitely is not, is called infinitesimal. If $$q$$ is a finite $$( q \cdot 1 )$$ amount of charge then using $$dq$$ would be an infinitesimal $$( q \cdot 1/\infty )$$ amount of charge. See

Likewise when a quantity becomes so large that a regular finite quantity becomes insignificant then we call it infinite. We would say that the mass of the ocean is infinite $$( M \cdot \infty )$$. But compared to the mass of the Milky Way galaxy our ocean is insignificant. So we would say the mass of the Galaxy is doubly infinite $$( M \cdot \infty^2 )$$.

Infinity and the infinitesimal are called (denoted $${}^*\mathbb{R}$$). Hyperreals behave, in every way, exactly like real numbers. For example, $$2 \cdot \infty$$ is exactly twice as big as $$\infty.$$ In reality, the mass of the ocean is a real number so it is hardly surprising that it behaves like one. See and

In ancient times infinity was called the "all".

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Groups and rings

 * Main articles:, , and

Addition and multiplication can be generalized in so many ways that mathematicians have created a whole system just to categorize them.



A is a set with a single  binary operation (usually,, addition. See ).
 * a + b = c

A is a magma where the addition is associative. See also
 * a + (b + c) = (a + b) + c

A is a semigroup with an additive identity element.
 * a + 0 = a

A is a monoid with additive inverse elements.
 * a + (-a) = 0

An is a group where the addition is commutative.
 * a + b = b + a

A is an abelian group that also has a second closed, associative, binary operation (usually, but not always, multiplication).
 * a * (b * c) = (a * b) * c
 * And these two operations satisfy a distribution law.
 * a(b + c) = ab + ac

A is a pseudo-ring that has a multiplicative identity
 * a * 1 = a

A is a ring where multiplication commutes, (e.g. )
 * a * b = b * a

A is a commutative ring where every element has a multiplicative inverse (and thus there is a multiplicative identity),
 * a * (1/a) = 1
 * The existence of a multiplicative inverse for every nonzero element automatically implies that there are no in a field
 * if ab=0 for some a≠0, then we must have b=0 (we call this having no zero-divisors).

$$\mathbb{Z}/n\mathbb{Z}$$ is the of $$\mathbb{Z}$$ by the ideal $$n\mathbb{Z}$$ containing all integers divisible by $4n + 3$.


 * Thus $$\mathbb{Z}/n\mathbb{Z}$$ is a field when $$n\mathbb{Z}$$ is a, that is, when $n$ is prime.

The of a  is the commutative subgroup of elements $c$ such that $n$ for every $x$. See also:.

The of a  is the commutative subring of elements $c$ such that $c+x = x+c$ for every $x$.

The of ring R, denoted char(R), is the number of times one must add the  to get the.



A is a group that is also a smooth differentiable manifold, in which the group operation is multiplication rather than addition. (Differentiation requires the ability to multiply and divide which is usually impossible with most groups.)

All non-zero elements are.


 * The $$A = \begin{pmatrix}

0 & 1 & 0\\   0 & 0 & 1\\     0 & 0 & 0  \end{pmatrix} $$ is nilpotent

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Numbers dont lie. (But they sure help)
From Mathematical fallacy:

$$ \begin{align} a &= b \\ a^2 &= ab \\ a^2 - b^2 &= ab - b^2 \\ (a - b)(a + b) &= b(a - b) \\ a + b &= b \\ b + b &= b \\ 2b &= b \\ 2 &= 1 \end{align} $$

The fallacy is in line 5: the progression from line 4 to line 5 involves division by a − b, which is zero since a = b. Since division by zero is undefined, the argument is invalid.

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Intervals

 * [-2,5[ or [-2,5) denotes the from -2 to 5, including -2 but excluding 5.
 * [3..7] denotes all integers from 3 to 7.
 * The set of all reals is unbounded at both ends.
 * An open interval does not include its endpoints.
 * is a property that generalizes the notion of a subset being closed and bounded.
 * The is the closed interval [0,1].  It is often denoted I.
 * The is a square whose sides have length 1.
 * Often, "the" unit square refers specifically to the square in the with corners at the four points (0, 0), (1, 0), (0, 1), and (1, 1).
 * The in the complex plane is the set of all complex numbers of absolute value less than one and is often denoted $$ \mathbb {D}$$

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Vectors

 * See also:, , , , and

The one dimensional number line can be generalized to a multidimensional thereby creating multidimensional math (i.e. ). See also

For sets A and B, the A × B is the set of all  (a, b) where a ∈ A and b ∈ B.


 * $$\mathbb{R}^3$$ is the Cartesian product $$\mathbb{R} \times \mathbb{R} \times \mathbb{R}.$$


 * $$\mathbb{R}^\infty = \mathbb{R}^\mathbb{N}$$


 * $$\mathbb{C}^3$$ is the $$\mathbb{C} \times \mathbb{C} \times \mathbb{C}$$ (See )

A is a  with  and  (multiplication of a vector and a  belonging to a ).




 * If $${\mathbf e_1}, {\mathbf e_2} , {\mathbf e_3} $$ are


 * and $${\mathbf u}, {\mathbf v} , {\mathbf x} $$ are arbitrary vectors


 * and are $$u_n, v_n , x_n$$ scalars belonging to a field then we can (and usually do) write:


 * $$\mathbf{u} = u_1 \mathbf{e_1} + u_2 \mathbf{e_2} + u_3 \mathbf{e_3} = \begin{bmatrix} u_1 & u_2 & u_3 \end{bmatrix}$$


 * $$\mathbf{v} = v_1 \mathbf{e_1} + v_2 \mathbf{e_2} + v_3 \mathbf{e_3} = \begin{bmatrix} v_1 & v_2 & v_3 \end{bmatrix}$$


 * $$\mathbf{x} = x_1 \mathbf{e_1} + x_2 \mathbf{e_2} + x_3 \mathbf{e_3} = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix}$$


 * See also:


 * A generalizes a vector space by allowing multiplication of a vector and a scalar belonging to a.

Coordinate systems define the length of vectors parallel to one of the axes but leave all other lengths undefined. This concept of "length" which only works for certain vectors is generalized as the "" which works for all vectors. The norm of vector $$\mathbf{v}$$ is denoted $$\|\mathbf{v}\|.$$ The double bars are used to avoid confusion with the absolute value of the function.


 * (called L1 norm. See . Sometimes called Lebesgue spaces. See also .) A circle in L1 space is shaped like a diamond.


 * $$\|\mathbf{v}\| = v_1 + v_2 + v_3$$




 * In the norm (called L2 norm) doesnt depend on the choice of coordinate system. As a result, rigid objects can rotate in Euclidean space. See proof of the  to the right. $cx = xc$ is the only  among $L^{2}$ spaces.


 * $$\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2}$$


 * In (See ) the  is


 * $$\|s\| = \sqrt{x^2 + y^2 + z^2 + (cti)^2}$$


 * In the most common norm of an n dimensional vector is obtained by treating it as though it were a regular real valued 2n dimensional vector in Euclidean space


 * $$\left\| \boldsymbol{z} \right\| = \sqrt{z_1 \bar z_1 + \cdots + z_n \bar z_n}$$


 * Infinity norm. (In this space a circle is shaped like a square.)


 * $$ \left\| \mathbf{x} \right\| _\infty := \max \left( \left| x_1 \right|, \ldots , \left| x_n \right| \right) .$$


 * A is a  that is also a  (there are no points missing from it).

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Spaces


Around 1735, Euler discovered the formula $$V - E + F = 2$$ relating the number of vertices, edges and faces of a convex polyhedron, and hence of a. No metric is required to prove this formula. The study and generalization of this formula is the origin of.

A topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence.



The metric is a function that defines a concept of distance between any two points. The distance from a point to itself is zero. The distance between two distinct points is positive.




 * style="width:20px"| 1.
 * style="width:250px"|$$d(x,y) \ge 0 $$
 * 2. || $$d(x,y) = 0 \quad$$ iff $$\quad x = y$$
 * 3. || $$d(x,z) \le d(x,y) + d(y, z)$$
 * 4. || $$d(x,y) = d(y,x) $$
 * }
 * 3. || $$d(x,z) \le d(x,y) + d(y, z)$$
 * 4. || $$d(x,y) = d(y,x) $$
 * }
 * }

A norm is the generalization to real vector spaces of the intuitive notion of distance in the real world. All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology (although the resulting metric spaces need not be the same).

A norm is a function that assigns a strictly positive length or size to each vector in a vector space—except for the zero vector, which is assigned a length of zero.
 * 1) $$\|\mathbf{v}\| ≥ 0$$
 * 2) $$\|\mathbf{v}\| = 0 \quad$$  iff  $$\quad \mathbf{v} = \mathbf{0}$$ (the zero vector)
 * 3) $$\|\mathbf{u} + \mathbf{v}\| ≤ \|\mathbf{u}\| + \|\mathbf{v}\| \quad$$  (The )
 * 4) $$\|\mathbf{v}\| = \|\mathbf{-v}\|$$
 * 5) $$\|a \cdot \mathbf{v}\| = |a| \cdot \|\mathbf{v}\|$$

A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector).

From List of vector spaces in mathematics:







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Multiplication of vectors
Multiplication can be generalized to allow for multiplication of vectors in 3 different ways:

Dot product
(a ): $$

\mathbf{u} \cdot \mathbf{v} =

\| \mathbf{u} \|\ \| \mathbf{v}\| \cos(\theta) =

u_1 v_1 + u_2 v_2 + u_3 v_3

$$


 * $$\mathbf{u}\cdot\mathbf{v} =

\begin{bmatrix}u_1

\mathbf{e_1} \\

u_2 \mathbf{e_2} \\

u_3 \mathbf{e_3}

\end{bmatrix}

\begin{bmatrix}v_1 \mathbf{e_1} & v_2 \mathbf{e_2} & v_3 \mathbf{e_3}

\end{bmatrix} =

\begin{bmatrix}u_1 v_1 + u_2 v_2 + u_3 v_3

\end{bmatrix}$$


 * Strangely, only parallel components multiply.


 * The dot product can be generalized to the $$\beta(\mathbf{u,v}) = u^T Av = scalar$$ where A is an (0,2) tensor. (For the dot product in Euclidean space A is the identity tensor. But in Minkowski space A is the ).


 * Two vectors are orthogonal if $$\beta(\mathbf{u,v}) = 0.$$


 * A bilinear form is symmetric if $$\beta(\mathbf{u,v}) = \beta(\mathbf{v,u})$$


 * Its associated is $$Q(\mathbf{x}) = \beta(\mathbf{x,x}).$$


 * In Euclidean space $$\|\mathbf{v}\|^2 = \mathbf{v}\cdot\mathbf{v}= Q(\mathbf{x}).$$


 * A nondegenerate bilinear form is one for which the associated matrix is invertible (its determinate is not zero)


 * $$\beta(\mathbf{u,v})=0 \,$$ for all v implies that u = 0.


 * The is a generalization of the dot product to complex vector space.


 * $$\langle u,v\rangle=\overline{\langle v,u\rangle}=u\cdot \bar{v}=\langle v \mid u\rangle$$ (See .)


 * The inner product can be generalized to a


 * A complex Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form $L^{p}$ such that $$h(w,z) = \overline{h(z, w)}.$$


 * A is a  $$\langle v \mid A u\rangle = \langle A v \mid u\rangle.$$ Often written as $$\langle v \mid A \mid u\rangle.$$


 * The curl operator, $$\nabla\times$$ is Hermitian.


 * A is an  that is also a.


 * The inner product of 2 functions $$f$$ and $$g$$ between $$a$$ and $$b$$ is


 * $$\langle f,g\rangle=\int\limits_a^b f\cdot\overline{g}\,dx$$


 * If this is equal to 0, the functions are said to be orthogonal on the interval. Unlike with vectors, this has no geometric significance but this definition is useful in . See below.

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Outer product
(a called a ): $$\mathbf{u} \otimes \mathbf{v}.$$


 * As one would expect, every component of one vector multipies with every component of the other vector.


 * For complex vectors, it is customary to use the conjugate transpose of v (denoted vH or v*):


 * $$\mathbf{u} \otimes \mathbf{v} = \mathbf{u} \mathbf{v}^\mathrm{H} = \mathbf{u} \mathbf{v}^*\ $$


 * Taking the dot product of u⊗v and any vector x (See ) causes the components of x not pointing in the direction of v to become zero. What remains is then rotated from v to u. Therefore an outer product rotates one component of a vector and causes all other components to become zero.


 * $$\mathbf{e}_1 \otimes \mathbf{e}_2 \cdot \mathbf{e}_2 = \mathbf{e}_1$$


 * To rotate a vector with 2 components you need the sum of at least 2 outer products (a bivector). But this is still not perfect. Any 3rd component not in the plane of rotation will become zero.


 * A true 3 dimensional rotation matrix can be constructed by summing three outer products. The first two sum to form a bivector. The third one rotates the axis of rotation zero degrees but is necessary to prevent that dimension from being squashed to nothing. $$\mathbf{e}_1 \otimes \mathbf{e}_2 - \mathbf{e}_2 \otimes \mathbf{e}_1 + \mathbf{e}_3 \otimes \mathbf{e}_3$$


 * The generalizes the.

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Geometric product
The will be explained in detail below.

Wedge product


(a simple ): $$\mathbf{u} \wedge \mathbf{v} = \mathbf{u} \otimes \mathbf{v} - \mathbf{v} \otimes \mathbf{u} = [\overline{\mathbf{u}}, \overline{\mathbf{v}}]$$


 * The wedge product of 2 vectors is equal to the minus the inner product as will be explained in detail below.


 * The wedge product is also called the  (sometimes mistakenly called the outer product).


 * The term "exterior" comes from the exterior product of two vectors not being a vector.


 * Just as a vector has length and direction so a bivector has an area and an orientation.


 * In three dimensions $$\mathbf{u} \wedge \mathbf{v}$$ is the of the  which is a . $$\overline{\mathbf{u} \wedge \mathbf{v}} = \mathbf{u} \times \mathbf{v}$$





\mathbf{a \wedge b \wedge c =

a \otimes b \otimes c -

a \otimes c \otimes b +

c \otimes a \otimes b -

c \otimes b \otimes a +

b \otimes c \otimes a -

b \otimes a \otimes c}

$$


 * }




 * The a∧b∧c is a trivector which is a 3rd degree tensor.


 * In 3 dimensions a trivector is a pseudoscalar so in 3 dimensions every trivector can be represented as a scalar times the unit trivector. See


 * $$\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c}) \cdot \mathbf{e}_1 \wedge \mathbf{e}_2 \wedge \mathbf{e}_3 $$


 * The of vector a is bivector ā:


 * $$\overline{\mathbf{a}} \quad\stackrel{\rm def}{=} \quad\begin{bmatrix}\,\,0&\!-a_3&\,\,\,a_2\\\,\,\,a_3&0&\!-a_1\\\!-a_2&\,\,a_1&\,\,0\end{bmatrix}$$

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Covectors
The Mississippi flows at about 3 km per hour. Km per hour has both direction and magnitude and is a vector.

The Mississippi flows downhill about one foot per km. Feet per km has direction and magnitude but is not a vector. Its a covector.

The difference between a vector and a covector becomes apparent when doing a changing units. If we measured in meters instead of km then 3 km per hour become 3000 meters per hour. The numerical value increases. Vectors are therefore contravariant.

But 1 Foot per km becomes 0.001 foot per meter. The numerical value decreases. Covectors are therefore covariant.

Tensors are more complicated. They can be part contravariant and part covariant.

A (1,1) Tensor is one part contravariant and one part covariant. It is totally unaffected by a change of units. It is these that we will study in the next section.

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Tensors

 * See also: and


 * External links: Review of Linear Algebra and High-Order Tensors



Just as a vector is a sum of unit vectors multiplied by constants so a tensor is a sum of unit dyadics ($$e_1 \otimes e_2$$) multiplied by constants. Each dyadic is associated with a certain plane segment having a certain orientation and magnitude. (But a dyadic is not the same thing as a .)

A simple tensor is a tensor that can be written as a product of tensors of the form $$T=a\otimes b\otimes\cdots\otimes d.$$ (See Outer Product above.) The rank of a tensor T is the minimum number of simple tensors that sum to T. A is a tensor of rank 2.

The order or degree of the tensor is the dimension of the tensor which is the total number of indices required to identify each component uniquely. A vector is a 1st-order tensor.

Complex numbers can be used to represent and but only in 2 dimensions.

, on the other hand, can be used in any number of dimensions to represent and perform rotations and other. See the image to the right.


 * Any is equivalent to a linear transformation followed by a  of the origin. (The  is always a fixed point for any linear transformation.) "Translation" is just a fancy word for "move".

Multiplying a tensor and a vector results in a new vector that can not only have a different magnitude but can even point in a completely different direction:



\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} a_1x+a_2y+a_3z\\ b_1x+b_2y+b_3z\\ c_1x+c_2y+c_3z\\ \end{bmatrix} $$

Some special cases:



\begin{bmatrix} {\color{green}a_1} & a_2 & a_3 \\ {\color{green}b_1} & b_2 & b_3 \\ {\color{green}c_1} & c_2 & c_3 \\ \end{bmatrix} \begin{bmatrix} 1\\   0\\    0\\  \end{bmatrix} = \begin{bmatrix} a_1\\ b_1\\ c_1\\ \end{bmatrix} $$



\begin{bmatrix} a_1 & {\color{green}a_2} & a_3 \\ b_1 & {\color{green}b_2} & b_3 \\ c_1 & {\color{green}c_2} & c_3 \\ \end{bmatrix} \begin{bmatrix} 0\\   1\\    0\\  \end{bmatrix} = \begin{bmatrix} a_2\\ b_2\\ c_2\\ \end{bmatrix} $$



\begin{bmatrix} a_1 & a_2 & {\color{green}a_3} \\ b_1 & b_2 & {\color{green}b_3} \\ c_1 & c_2 & {\color{green}c_3} \\ \end{bmatrix} \begin{bmatrix} 0\\   0\\    1\\  \end{bmatrix} = \begin{bmatrix} a_3\\ b_3\\ c_3\\ \end{bmatrix} $$

One can also multiply a tensor with another tensor. Each column of the second tensor is transformed exactly as a vector would be.



\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\   0 & 1 & 0 \\    0 & 0 & 1 \\  \end{bmatrix}  = \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{bmatrix} $$

And we can also switch things around using a. (See also ):



\begin{bmatrix} 1 & 0 & 0 \\   0 & 0 & 1 \\    0 & 1 & 0 \\  \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} x\\ z\\ y\\ \end{bmatrix} $$



\begin{bmatrix} 0 & 0 & 1 \\   0 & 1 & 0 \\    1 & 0 & 0 \\  \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} z\\ y\\ x\\ \end{bmatrix} $$



\begin{bmatrix} 0 & 1 & 0 \\   1 & 0 & 0 \\    0 & 0 & 1 \\  \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} y\\ x\\ z\\ \end{bmatrix} $$

Matrices do not in general commute:



\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \begin{pmatrix} 0 & 1 \\  1 & 0  \end{pmatrix} = \begin{pmatrix} 0 & a \\ b & 0 \end{pmatrix} $$


 * but



\begin{pmatrix} 0 & 1 \\  1 & 0  \end{pmatrix} \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}  = \begin{pmatrix} 0 & b \\ a & 0 \end{pmatrix} $$

The of a matrix is the area or volume of the n-dimensional parallelepiped spanned by its column (or row) vectors and is frequently useful.


 * $$|A|=\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc$$

Matrices do have zero divisors:



A= \begin{pmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\0&0&0&0\end{pmatrix}, \quad

A^2= \begin{pmatrix}0&0&1&0\\0&0&0&1\\0&0&0&0\\0&0&0&0\end{pmatrix}, \quad

A^3= \begin{pmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}, \quad

A^4=0$$

Normal matrices
A :



\begin{bmatrix} 6 & 0 & 0   \\ 0 & -7i & 0   \\ 0 & 0 & 1+9i \end{bmatrix}$$


 * The determinate of a diagonal matrix:



\begin{vmatrix} a&0&0\\ 0&b&0\\ 0&0&c \end{vmatrix} =abc $$

A superdiagonal entry is one that is directly above and to the right of the main diagonal. A subdiagonal entry is one that is directly below and to the left of the main diagonal. The eigenvalues of diag(λ1, ..., λn) are λ1, ..., λn with associated eigenvectors of e1, ..., en.

A is a result about when a matrix can be diagonalized. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations.

A matrix is normal if and only if it is.

Change of basis
An n-by-n square matrix A is invertible if there exists an n-by-n square matrix A-1 such that


 * $$\mathbf{AA^{-1}} = \mathbf{A^{-1}A} = \mathbf{I}_n \ $$

A matrix is invertible if and only if its determinant is non-zero.



\begin{align} \frac{|\mathbf{A}|}{|\mathbf{A}|} & = 1 \\ \end{align} $$
 * \mathbf{AA^{-1}}| & = |\mathbf{I}_n| \ \\
 * \mathbf{A}| |\mathbf{A^{-1}}| & = 1 \\
 * \mathbf{A}| & \neq 0

The standard basis for $$R^2$$ would be:


 * $$E_2 = \left\{ \begin{pmatrix} 1\\ 0\end{pmatrix}, \begin{pmatrix} 0\\ 1\end{pmatrix} \right\}$$

Given a matrix $$M$$ whose columns are the vectors of the new basis of the space (new basis matrix), the new coordinates for a column vector $$v$$ are given by the matrix product $$M^{-1}v$$.

From Matrix similarity:

Given a linear transformation:


 * $$y = Tx$$,

it can be the case that a change of basis can result in a simpler form of the same transformation.


 * $$y' = Sx'$$,


 * where $x'$ and $y'$ are in the new basis.


 * $$x' = P x$$
 * $$y' = P y$$


 * and $P$ is the change-of-basis matrix.

To derive $T$ in terms of the simpler matrix, we use:


 * $$\begin{align}

y' &= Sx' \\ Py &= SPx \\ P^{-1}Py &= P^{-1}SPx \\ y &= \left(P^{-1}SP\right)x = Tx \end{align}$$

Thus, the matrix in the original basis is given by


 * $$T = P^{-1}SP$$


 * Therefore


 * $$S = PTP^{-1}$$

From Matrix similarity

Two n-by-n matrices $A$ and $B$ are called similar if


 * $$B = P^{-1} A P $$

for some invertible n-by-n matrix $P$.

A transformation $h : V × V → C$ is called a similarity transformation or conjugation of the matrix $A$. In the, similarity is therefore the same as , and similar matrices are also called conjugate.

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Linear groups
A of order n is an n-by-n matrix. Any two square matrices of the same order can be added and multiplied. A matrix is invertible if and only if its determinant is nonzero.

GLn(F) or GL(n, F), or simply GL(n) is the of n×n invertible matrices with entries from the field F. The group operation is. The group GL(n, F) and its subgroups are often called linear groups or matrix groups.


 * SL(n, F) or SLn(F), is the of GL(n, F) consisting of matrices with a  of 1.


 * U(n), the Unitary group of degree n is the of n × n . The group operation is . The determinant of a unitary matrix is a complex number with norm 1.


 * SU(n), the special unitary group of degree $A ↦ P^{−1}AP$, is the of $n$  with  1.

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Symmetry groups

 * boosts, rotations, translations


 * boosts, rotations


 * The set of all boosts, however, does not form a subgroup, since composing two boosts does not, in general, result in another boost. (Rather, a pair of non-colinear boosts is equivalent to a boost and a rotation, and this relates to Thomas rotation.)

Aff(n,K): the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.


 * E(n): rotations, reflections, and translations.


 * O(n): rotations, reflections


 * SO(n): rotations


 * so(3) is the Lie algebra of $n×n$ and consists of all $SO(3)$ matrices.

Clifford group: The set of invertible elements x such that for all v in V $$x v \alpha(x)^{-1}\in V .$$ The  Q is defined on the Clifford group by $$Q(x) = x^\mathrm{t}x.$$


 * PinV(K): The subgroup of elements of spinor norm 1. Maps 2-to-1 to the orthogonal group


 * SpinV(K): The subgroup of elements of Dickson invariant 0 in PinV(K). When the characteristic is not 2, these are the elements of determinant 1. Maps 2-to-1 to the special orthogonal group. Elements of the spin group act as linear transformations on the space of spinors

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Rotations
In 4 spatial dimensions a rigid object can.




 * See also:, , , , , , , , , , , ,


 * From Rotation group SO(3):

Consider the solid ball in R3 of radius π. For every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. The two rotations through π and through −π are the same. So we (or "glue together")  on the surface of the ball.

The ball with antipodal surface points identified is a, and this manifold is  to the rotation group. It is also diffeomorphic to the RP3, so the latter can also serve as a topological model for the rotation group.

These identifications illustrate that SO(3) is but not. As to the latter, consider the path running from the "north pole" straight through the interior down to the south pole. This is a closed loop, since the north pole and the south pole are identified. This loop cannot be shrunk to a point, since no matter how you deform the loop, the start and end point have to remain antipodal, or else the loop will "break open". (In other words one full rotation is not equivalent to doing nothing.)



Surprisingly, if you run through the path twice, i.e., run from north pole down to south pole, jump back to the north pole (using the fact that north and south poles are identified), and then again run from north pole down to south pole, so that φ runs from 0 to 4π, you get a closed loop which can be shrunk to a single point: first move the paths continuously to the ball's surface, still connecting north pole to south pole twice. The second half of the path can then be mirrored over to the antipodal side without changing the path at all. Now we have an ordinary closed loop on the surface of the ball, connecting the north pole to itself along a great circle. This circle can be shrunk to the north pole without problems. The and similar tricks demonstrate this practically.

The same argument can be performed in general, and it shows that the of SO(3) is  of order 2. In physics applications, the non-triviality of the fundamental group allows for the existence of objects known as, and is an important tool in the development of the.

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Matrix representations

 * See also:, ,

Real numbers
If a vector is multiplied with the the $$I$$ then the vector is completely unchanged:


 * $$ I \cdot v =

\begin{bmatrix} 1 & 0 & 0 \\   0 & 1 & 0 \\    0 & 0 & 1 \\  \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = 1 \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} $$

And if $$A=a \cdot I$$ then


 * $$ A \cdot v =

\begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \\ \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = a \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} a\cdot x\\ a \cdot y\\ a\cdot z\\ \end{bmatrix} $$

Therefore $$A=a \cdot I$$ can be thought of as the matrix form of the scalar a. The scalar matrices are the center of the algebra of matrices.


 * $$ A + B =

\begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \\ \end{bmatrix} + \begin{bmatrix} b & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & b \\ \end{bmatrix} = \begin{bmatrix} a + b & 0 & 0 \\ 0 & a + b & 0 \\ 0 & 0 & a + b \\ \end{bmatrix} $$


 * $$ A \cdot B =

\begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \\ \end{bmatrix} \begin{bmatrix} b & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & b \\ \end{bmatrix} = \begin{bmatrix} ab & 0 & 0 \\ 0 & ab & 0 \\ 0 & 0 & ab \\ \end{bmatrix} $$


 * $$ 1/A =

\begin{bmatrix} 1/a & 0 & 0 \\ 0 & 1/a & 0 \\ 0 & 0 & 1/a \\ \end{bmatrix} $$


 * $$ |A| =

\begin{vmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{vmatrix} = a^3 $$


 * $$ A^B =

\begin{bmatrix} a^b & 0 & 0 \\ 0 & a^b & 0 \\ 0 & 0 & a^b \\ \end{bmatrix} $$


 * $$e^A=

\begin{bmatrix} e^a & 0 & 0 \\ 0 & e^a & 0 \\ 0 & 0 & e^a \\ \end{bmatrix} $$.


 * $$\ln A=

\begin{bmatrix} \ln a & 0 & 0 \\ 0 & \ln a & 0 \\ 0 & 0 & \ln a \\ \end{bmatrix} $$.

(Note: Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.)

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Complex numbers
Complex numbers can also be in such a way that complex multiplication corresponds perfectly to matrix multiplication:



\begin{align} (a+ib)(c+id) &= \begin{bmatrix} a & b \\ -b & a \end{bmatrix} \begin{bmatrix} c & d \\ -d & c \end{bmatrix} \\ &= \begin{bmatrix} ac-bd & ad+bc \\ -(ad+bc) & ac-bd \end{bmatrix} \end{align} $$



\begin{align} (i)(i) &= \begin{bmatrix} 0 & 1 \\    -1  & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\    -1  & 0 \end{bmatrix} \\ &= \begin{bmatrix} -1 & 0 \\    0  & -1 \end{bmatrix} \\ &= -I \end{align} $$

The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the origin computed using the Pythagorean theorem.


 * $$ |z|^2 =

\begin{vmatrix} a & -b \\ b & a \end{vmatrix} = a^2 + b^2. $$

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Quaternions
There are at least two ways of representing quaternions as in such a way that quaternion addition and multiplication correspond to matrix addition and.

Using 2&thinsp;×&thinsp;2 complex matrices, the quaternion a + bi + cj + dk can be represented as



\begin{bmatrix} z_1 & z_2 \\ -\overline{z_2} & \overline{z_1} \end{bmatrix} = \begin{bmatrix} a+bi & c+di \\ -(c-di) & a-bi \end{bmatrix}= a \begin{bmatrix} 1 & 0 \\  0 & 1  \\ \end{bmatrix} + b \begin{bmatrix} i & 0 \\ 0 & -i \\ \end{bmatrix} + c \begin{bmatrix} 0 & 1 \\  -1 & 0  \\ \end{bmatrix} + d \begin{bmatrix} 0 & i \\ i & 0 \\ \end{bmatrix}. $$

Multiplying any two Pauli matrices always yields a quaternion unit matrix. See Isomorphism to quaternions below.

By replacing each 0, 1, and i with its 2 × 2 matrix representation that same quaternion can be written as a 4 × 4 real matrix:



$$\begin{bmatrix} a & b & c & d \\ -b & a & -d & c \\ -c & d & a & -b \\ -d & -c & b & a \end{bmatrix}= a \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} + b \begin{bmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{bmatrix} + c \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{bmatrix} + d \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{bmatrix}. $$
 * }

Therefore:



\begin{align} b \cdot i &= b \cdot (e_1 \wedge e_2 + e_4 \wedge e_3)\\ c \cdot j &= c \cdot (e_1 \wedge e_3 + e_2 \wedge e_4)\\ d \cdot k &= d \cdot (e_1 \wedge e_4 + e_3 \wedge e_2)\\ \end{align} $$

However, the representation of quaternions in M(4,ℝ) is not unique. In fact, there exist 48 distinct representations of this form. Each 4x4 matrix representation of quaternions corresponds to a multiplication table of unit quaternions. See Quaternion.

The obvious way of representing quaternions with 3&thinsp;×&thinsp;3 real matrices does not work because:


 * $$ b

\begin{bmatrix} 0 & -1 & 0 \\  1 & 0 & 0  \\ 0 & 0 & 0  \end{bmatrix} \cdot c \begin{bmatrix} 0 & 0 & -1 \\  0 & 0 & 0  \\ 1 & 0 & 0  \end{bmatrix} \neq d \begin{bmatrix} 0 & 0 & 0 \\  0 & 0 & -1  \\ 0 & {\color{red}1} & 0 \end{bmatrix}. $$

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Euclidean

 * See also:

Unfortunately the matrix representation of a vector is not so obvious. First we must decide what properties the matrix should have. To see consider the square of a single vector:




 * $$Q(c) = c^2 = \langle c, c \rangle = (a\mathbf{e_1} + b\mathbf{e_2})^2$$


 * $$Q(c) = aa\mathbf{e_1}\mathbf{e_1} + bb\mathbf{e_2}\mathbf{e_2} + ab\mathbf{e_1}\mathbf{e_2} + ba\mathbf{e_2}\mathbf{e_1}$$


 * $$Q(c) = a^2\mathbf{e_1}\mathbf{e_1} + b^2\mathbf{e_2}\mathbf{e_2} + ab(\mathbf{e_1}\mathbf{e_2} + \mathbf{e_2}\mathbf{e_1})$$

From the Pythagorean theorem we know that:


 * $$c^2 = a^2 + b^2 + ab(0) = Scalar$$

So we know that


 * $$e_1^2 = e_2^2 = 1$$


 * $$e_1 e_2 = -e_2 e_1 $$

This particular Clifford algebra is known as Cl2,0. The subscript 2 indicates that the 2 basis vectors are square roots of +1. See. If we had used $$c^2 = -a^2 -b^2$$ then the result would have been Cl0,2.

The set of 3 matrices in 3 dimensions that have these properties are called. The algebra generated by the three Pauli matrices is isomorphic to the Clifford algebra of $3 × 3$.


 * From Pauli matrices

The Pauli matrices are a set of three $ℝ^{3}$  which are  and. They are

\begin{align} \sigma_1 = \sigma_x &= \begin{pmatrix} 0&1\\     1&0    \end{pmatrix} \\ \sigma_2 = \sigma_y &= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} \\ \sigma_3 = \sigma_z &= \begin{pmatrix} 1&0\\     0&-1    \end{pmatrix} \,. \end{align}$$

Squaring a Pauli matrix results in a "scalar":


 * $$\sigma_1^2 = \sigma_2^2 = \sigma_3^2 =

\begin{pmatrix} 1&0\\     0&1    \end{pmatrix} = \sigma_0 = I$$

Do NOT confuse this scalar with the vectors above. It may look similar to the Pauli matrices but it is not the matrix representation of a vector. It is the matrix representation of a scalar. Scalars are totally different from vectors and the matrix representations of scalars are totally different from the matrix representations of vectors. They are NOT the same.

Multiplication is :


 * $$\sigma_1 \sigma_2 = - \sigma_2  \sigma_1$$


 * $$\sigma_2 \sigma_3 = - \sigma_3  \sigma_2$$


 * $$\sigma_3 \sigma_1 = - \sigma_1  \sigma_3$$

And


 * $$\sigma_1 \sigma_2 \sigma_3 = \begin{pmatrix} i&0\\0&i\end{pmatrix} = i$$

Exponential of a Pauli vector which is analogous to Euler's formula, extended to quaternions:


 * $$e^{i a(\hat{n} \cdot \vec{\sigma})} = I\cos{a} + i (\hat{n} \cdot \vec{\sigma}) \sin{a} $$

relations:
 * $$\begin{align}

\left[\sigma_1, \sigma_2\right] &= 2i\sigma_3 \, \\ \left[\sigma_2, \sigma_3\right] &= 2i\sigma_1 \, \\ \left[\sigma_3, \sigma_1\right] &= 2i\sigma_2 \, \\ \left[\sigma_1, \sigma_1\right] &= 0\, \\ \end{align}$$

relations:


 * $$\begin{align}

\left\{\sigma_1, \sigma_1\right\} &= 2I\, \\ \left\{\sigma_1, \sigma_2\right\} &= 0\,.\\ \end{align}$$

Adding the commutator ($$ab - ba$$) to the anticommutator ($$ab + ba$$) gives the general formula for multiplying any 2 arbitrary "vectors" (or rather their matrix representations):


 * $$(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma}) = (\vec{a} \cdot \vec{b}) \, I + i ( \vec{a} \times \vec{b} )\cdot \vec{\sigma}$$

If $$ i $$ is identified with the pseudoscalar $$ \sigma_x \sigma_y \sigma_z $$ then the right hand side becomes $$ a \cdot b + a \wedge b $$ which is also the definition for the of two vectors in. The geometric product of two vectors is a.

For any 2 arbitrary vectors:


 * $$fd = force*distance$$


 * $$fd = (f_1\mathbf{e_1} + f_2\mathbf{e_2})(d_1\mathbf{e_1} + d_2\mathbf{e_2})$$


 * $$fd = f_1d_1\mathbf{e_1}\mathbf{e_1} + f_2d_2\mathbf{e_2}\mathbf{e_2} + f_1d_2\mathbf{e_1}\mathbf{e_2} + f_2d_1\mathbf{e_2}\mathbf{e_1}$$


 * $$fd = f_1d_1\mathbf{e_1}\mathbf{e_1} + f_2d_2\mathbf{e_2}\mathbf{e_2} + f_1d_2\mathbf{e_1}\mathbf{e_2} - f_2d_1\mathbf{e_1}\mathbf{e_2}$$


 * $$fd = f_1d_1\mathbf{e_1}\mathbf{e_1} + f_2d_2\mathbf{e_2}\mathbf{e_2} + (f_1d_2 - f_2d_1)\mathbf{e_1}\mathbf{e_2}$$

Applying the rules of Clifford algebra we get:


 * $$fd = f_1d_1 + f_2d_2 + (f_1d_2 - f_2d_1)\mathbf{e_1} \wedge \mathbf{e_2}$$


 * $$fd = Energy + Torque$$


 * $$fd = {\color{red} f \cdot d} + {\color{blue} f \wedge d}$$


 * $$fd = {\color{red} Scalar} + {\color{blue} Bivector} = Multivector$$

Further reading:, and

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Pseudo-Euclidean

 * See also:


 * From Gamma matrices

Gamma , $$ \{ \gamma^0, \gamma^1, \gamma^2, \gamma^3 \} $$, also known as the Dirac matrices, are a set of $2 × 2$ conventional matrices with specific relations that ensure they  a matrix representation of the  Cℓ1,3(R). One gamma matrix squares to 1 times the and three gamma matrices square to -1 times the identity matrix.



(\gamma^0)^2 = I $$



(\gamma^1)^2 = (\gamma^2)^2 = (\gamma^3)^2 = -I $$

The defining property for the gamma matrices to generate a is the anticommutation relation


 * $$\displaystyle\{ \gamma^\mu, \gamma^\nu \} = \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu \nu} I_4 $$

where $$\{, \}$$ is the , $$\eta^{\mu \nu}$$ is the with signature (+ − − −) and $$I_4$$ is the 4 × 4 identity matrix.

When interpreted as the matrices of the action of a set of orthogonal basis vectors for vectors in, the column vectors on which the matrices act become a space of , on which the Clifford algebra of  acts. This in turn makes it possible to represent infinitesimal and. Spinors facilitate spacetime computations in general, and in particular are fundamental to the for relativistic spin-½ particles.

In, the four gamma matrices are


 * $$\begin{align}

\gamma^0 &= \begin{pmatrix} 1 & 0 & 0 &  0 \\  0 & 1 &  0 &  0 \\   0 & 0 &  -1 &  0 \\  0 & 0 &  0 & -1 \end{pmatrix},\quad& \gamma^1 &= \begin{pmatrix} 0 & 0 & 0 & 1 \\   0 &  0 & 1 & 0 \\   0 & -1 & 0 & 0 \\  -1 &  0 & 0 & 0 \end{pmatrix} \\

\gamma^2 &= \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{pmatrix},\quad& \gamma^3 &= \begin{pmatrix} 0 & 0 & 1 & 0 \\   0 & 0 & 0 & -1 \\  -1 & 0 & 0 &  0 \\   0 & 1 & 0 &  0 \end{pmatrix}. \end{align}$$

$$\gamma^0$$ is the time-like matrix and the other three are space-like matrices.



(\gamma^0)^2 = \begin{pmatrix} 1 & 0 & 0 &  0 \\  0 & 1 &  0 &  0 \\   0 & 0 &  1 &  0 \\  0 & 0 &  0 & 1 \end{pmatrix} $$



(\gamma^1)^2 = (\gamma^2)^2 = (\gamma^3)^2 = \begin{pmatrix} -1 & 0 & 0 &  0 \\  0 & -1 &  0 &  0 \\   0 & 0 &  -1 &  0 \\  0 & 0 &  0 & -1 \end{pmatrix} $$

The matrices are also sometimes written using the 2×2, $$I_2$$, and the.

The gamma matrices we have written so far are appropriate for acting on written in the Dirac basis; in fact, the Dirac basis is defined by these matrices. To summarize, in the Dirac basis:


 * $$\gamma^0 = \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix},\quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix}.$$

Another common choice is the Weyl or chiral basis, in which $$\gamma^k$$ remains the same but $$\gamma^0$$ is different, and so $$\gamma^5$$ is also different, and diagonal,


 * $$\gamma^0 = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix},\quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} -I_2 & 0 \\ 0 & I_2 \end{pmatrix},$$

Further reading: Quan­tum Me­chan­ics for En­gi­neers and How (not) to teach Lorentz covariance of the Dirac equation

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Multivectors

 * See also:

External links:
 * A brief introduction to geometric algebra
 * A brief introduction to Clifford algebra
 * The Construction of Spinors in Geometric Algebra
 * Functions of Multivector Variables
 * Clifford Algebra Representations

Rotors

 * See also:


 * From Geometric algebra

The inverse of a vector is:


 * $$ v^{-1} = \frac{1}{v} = \frac{v}{vv} = \frac{v}{v \cdot v + v \wedge v} = \frac{v}{v \cdot v}$$

The projection of $$v$$ onto $$a$$ (or the parallel part) is


 * $$ v_{\| a} = (v \cdot a)a^{-1} $$

and the rejection of $$v$$ from $$a$$ (or the orthogonal part) is


 * $$ v_{\perp a} = v - v_{\| a} = (v\wedge a)a^{-1} .$$

The reflection $$v'$$ of a vector $$v$$ along a vector $$a$$, or equivalently across the hyperplane orthogonal to $$a$$, is the same as negating the component of a vector parallel to $$a$$. The result of the reflection will be

If a is a unit vector then $$a^{-1}=\frac{a}{1} = a$$ and therefore $$v' = -ava $$

$$-ava$$ is called the sandwich product which is called a double-sided product.

If we have a product of vectors $$R = a_1a_2 \cdots a_r$$ then we denote the reverse as


 * $$R^\dagger = (a_1a_2\cdots a_r)^\dagger = a_r\cdots a_2 a_1.$$

Any rotation is equivalent to 2 reflections.


 * $$v'' = bv'b = bavab = RvR^\dagger$$

R is called a Rotor


 * $$R = ba = b \cdot a + b \wedge a = Scalar + Bivector = Multivector$$

If a and b are unit vectors then the rotor is automatically normalised:


 * $$RR^\dagger = R^\dagger R=1 .$$

2 rotations becomes:


 * $$R_2R_1MR_1^\dagger R_2^\dagger$$

R2R1 represents Rotor R1 rotated by Rotor R2. This would be called a single-sided transformation. (R2R1R2 would be double-sided.) Therefore rotors do not transform double-sided the same way that other objects do. They transform single-sided.

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Quaternions
The square root of the product of a quaternion with its conjugate is called its :


 * $$\lVert q \rVert = \sqrt{qq^*} = \sqrt{q^*q} = \sqrt{a^2 + b^2 + c^2 + d^2}$$

A unit quaternion is a quaternion of norm one. Unit quaternions, also known as, provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions.


 * From Quaternions and spatial rotation

Every nonzero quaternion has a multiplicative inverse


 * $$(a+bi+cj+dk)^{-1} = \frac{1}{a^2+b^2+c^2+d^2}\,(a-bi-cj-dk).$$

Thus quaternions form a.

The inverse of a unit quaternion is obtained simply by changing the sign of its imaginary components.

A such as $4 × 4$ or ${ e_{0}, e_{1}, e_{2}, e_{3} }$ can be rewritten as $v$ or $(v^{0}, v^{1}, v^{2}, v^{3})$, where $v = v^{μ}e_{μ}$, $v^{0}$, $v$ are unit vectors representing the three. A rotation through an angle of $v$ around the axis defined by a unit vector


 * $$\vec{u} = (u_x, u_y, u_z) = 0 + u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k}$$

can be represented by a quaternion. This can be done using an of :


 * $$ \mathbf{q} = e^{\frac{\theta}{2}{(0 + u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k})}} = \cos \frac{\theta}{2} + (0 + u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k}) \sin \frac{\theta}{2}$$

It can be shown that the desired rotation can be applied to an ordinary vector $$\mathbf{p} = (p_x, p_y, p_z) = 0 + p_x\mathbf{i} + p_y\mathbf{j} + p_z\mathbf{k}$$ in 3-dimensional space, considered as a quaternion with a real coordinate equal to zero, by evaluating the conjugation of $4$ by $v$:


 * $$\mathbf{p'} = \mathbf{q} \mathbf{p} \mathbf{q}^{-1}$$

using the

The conjugate of a product of two quaternions is the product of the conjugates in the reverse order.

Conjugation by the product of two quaternions is the composition of conjugations by these quaternions: If $3$ and $v = (v_{1}, v_{2}, v_{3})$ are unit quaternions, then rotation (conjugation) by $v$ is


 * $$\mathbf{p q} \vec{v} (\mathbf{p q})^{-1} = \mathbf{p q} \vec{v} \mathbf{q}^{-1} \mathbf{p}^{-1} = \mathbf{p} (\mathbf{q} \vec{v} \mathbf{q}^{-1}) \mathbf{p}^{-1}$$,

which is the same as rotating (conjugating) by $w$ and then by $η(v, v)$. The scalar component of the result is necessarily zero.

The imaginary part $$b\mathbf{i} + c\mathbf{j} + d\mathbf{k}$$ of a quaternion behaves like a vector $$\vec{v} = (b,c,d)$$ in three dimension vector space, and the real part $η$ behaves like a in $(0,2)$. When quaternions are used in geometry, it is more convenient to define them as :


 * $$a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} = a + \vec{v}.$$

When multiplying the vector/imaginary parts, in place of the rules $u, v$ we have the quaternion multiplication rule:
 * $$\vec{v} \vec{w} = \vec{v} \times \vec{w} - \vec{v} \cdot \vec{w},$$

From these rules it follows immediately that :


 * $$(s + \vec{v}) (t + \vec{w}) = (s t - \vec{v} \cdot \vec{w}) + (s \vec{w} + t \vec{v} + \vec{v} \times \vec{w}).$$

It is important to note, however, that the vector part of a quaternion is, in truth, an "axial" vector or "pseudovector", not an ordinary or "polar" vector.


 * From Quaternion:

the reflection of a vector r in a plane perpendicular to a unit vector w can be written:


 * $$r^{\prime} = - w\, r\, w.$$

Two reflections make a rotation by an angle twice the angle between the two reflection planes, so


 * $$v^{\prime\prime} = \sigma_2 \sigma_1 \, v \, \sigma_1 \sigma_2 $$

corresponds to a rotation of 180° in the plane containing σ1 and σ2.

This is very similar to the corresponding quaternion formula,


 * $$v^{\prime\prime} = -\mathbf{k}\, v\, \mathbf{k}. $$

In fact, the two are identical, if we make the identification


 * $$\mathbf{k} = \sigma_2 \sigma_1, \mathbf{i} = \sigma_3 \sigma_2, \mathbf{j} = \sigma_1 \sigma_3$$

and it is straightforward to confirm that this preserves the Hamilton relations


 * $$\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{i} \mathbf{j} \mathbf{k} = -1.$$

In this picture, quaternions correspond not to vectors but to – quantities with magnitude and orientations associated with particular 2D planes rather than 1D directions. The relation to becomes clearer, too: in 2D, with two vector directions σ1 and σ2, there is only one bivector basis element σ1σ2, so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements σ1σ2, σ2σ3, σ3σ1, so three imaginaries.

The usefulness of quaternions for geometrical computations can be generalised to other dimensions, by identifying the quaternions as the even part Cℓ+3,0(R) of the Cℓ3,0(R).

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Spinors

 * See also:

External link:An introduction to spinors

Spinors may be regarded as non-normalised rotors which transform single-sided.

Note: The (real) in three-dimensions are quaternions, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication.

A spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360°. This property characterizes spinors.


 * From Orientation entanglement

In three dimensions...the  is not. Mathematically, one can tackle this problem by exhibiting the   SU(2), which is also the  in three  dimensions, as a  of SO(3).

$M$ is the following group,


 * $$ \mathrm{SU}(2) = \left \{ \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta & \overline{\alpha} \end{pmatrix}: \ \ \alpha,\beta\in\mathbf{C}, |\alpha|^2 + |\beta|^2 = 1\right \} ~,$$

where the overline denotes.

For comparison: Using 2&thinsp;×&thinsp;2 complex matrices, the quaternion a + bi + cj + dk can be represented as



\begin{bmatrix} a+bi & c+di \\ -(c-di) & a-bi \end{bmatrix}.$$

If X = (x1,x2,x3) is a vector in R3, then we identify X with the 2 &times; 2 matrix with complex entries


 * $$X=\left(\begin{matrix}x_1&x_2-ix_3\\x_2+ix_3&-x_1\end{matrix}\right)$$

Note that &minus;det(X) gives the square of the Euclidean length of X regarded as a vector, and that X is a, or better, trace-zero.

The unitary group acts on X via


 * $$X\mapsto MXM^+$$

where M ∈ SU(2). Note that, since M is unitary,


 * $$\det(MXM^+) = \det(X)$$, and
 * $$MXM^+$$ is trace-zero Hermitian.

Hence SU(2) acts via rotation on the vectors X. Conversely, since any which sends trace-zero Hermitian matrices to trace-zero Hermitian matrices must be unitary, it follows that every rotation also lifts to SU(2). However, each rotation is obtained from a pair of elements M and &minus;M of SU(2). Hence SU(2) is a double-cover of SO(3). Furthermore, SU(2) is easily seen to be itself simply connected by realizing it as the group of unit, a space to the.

A unit quaternion has the cosine of half the rotation angle as its scalar part and the sine of half the rotation angle multiplying a unit vector along some rotation axis (here assumed fixed) as its pseudovector (or axial vector) part. If the initial orientation of a rigid body (with unentangled connections to its fixed surroundings) is identified with a unit quaternion having a zero pseudovector part and +1 for the scalar part, then after one complete rotation (2pi rad) the pseudovector part returns to zero and the scalar part has become -1 (entangled). After two complete rotations (4pi rad) the pseudovector part again returns to zero and the scalar part returns to +1 (unentangled), completing the cycle.


 * From Spinors in three dimensions

The association of a spinor with a 2×2 complex was formulated by Élie Cartan.

In detail, given a vector x = (x1, x2, x3) of real (or complex) numbers, one can associate the complex matrix
 * $$\vec{x} \rightarrow X \ =\left(\begin{matrix}x_3&x_1-ix_2\\x_1+ix_2&-x_3\end{matrix}\right).$$

Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3-space:
 * det X = – (length x)2.
 * X 2 = (length x)2I, where I is the identity matrix.
 * $$\frac{1}{2}(XY+YX)=({\bold x}\cdot{\bold y})I$$
 * $$\frac{1}{2}(XY-YX)=iZ$$ where Z is the matrix associated to the cross product z = x &times; y.
 * If u is a unit vector, then −UXU is the matrix associated to the vector obtained from x by reflection in the plane orthogonal to u.
 * It is an elementary fact from that any rotation in 3-space factors as a composition of two reflections.  (Similarly, any orientation reversing orthogonal transformation is either a reflection or the product of three reflections.)  Thus if R is a rotation, decomposing as the reflection in the plane perpendicular to a unit vector u1 followed by the plane perpendicular to u2, then the matrix U2U1XU1U2 represents the rotation of the vector x through R.

Having effectively encoded all of the rotational linear geometry of 3-space into a set of complex 2&times;2 matrices, it is natural to ask what role, if any, the 2&times;1 matrices (i.e., the ) play. Provisionally, a spinor is a column vector
 * $$\xi=\left[\begin{matrix}\xi_1\\\xi_2\end{matrix}\right],$$ with complex entries ξ1 and ξ2.

The space of spinors is evidently acted upon by complex 2&times;2 matrices. Furthermore, the product of two reflections in a given pair of unit vectors defines a 2&times;2 matrix whose action on euclidean vectors is a rotation, so there is an action of rotations on spinors.

Often, the first example of spinors that a student of physics encounters are the 2&times;1 spinors used in Pauli's theory of electron spin. The are a vector of three 2&times;2 that are used as.

Given a in 3 dimensions, for example (a, b, c), one takes a with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector.

The of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector.

Example: u = (0.8, -0.6, 0) is a unit vector. Dotting this with the Pauli spin matrices gives the matrix:



S_u = (0.8,-0.6,0.0)\cdot \vec{\sigma}=0.8 \sigma_{1}-0.6\sigma_{2}+0.0\sigma_{3} = \begin{bmatrix} 0.0 & 0.8+0.6i \\ 0.8-0.6i & 0.0 \end{bmatrix} $$

The eigenvectors may be found by the usual methods of, but a convenient trick is to note that a Pauli spin matrix is an, that is, the squareof the above matrix is the identity matrix.

Thus a (matrix) solution to the eigenvector problem with eigenvalues of ±1 is simply 1 ± Su. That is,



S_u (1\pm S_u) = \pm 1 (1 \pm S_u) $$

One can then choose either of the columns of the eigenvector matrix as the vector solution, provided that the column chosen is not zero. Taking the first column of the above, eigenvector solutions for the two eigenvalues are:



\begin{bmatrix} 1.0+ (0.0)\\ 0.0 +(0.8-0.6i) \end{bmatrix}, \begin{bmatrix} 1.0- (0.0)\\ 0.0-(0.8-0.6i) \end{bmatrix} $$

The trick used to find the eigenvectors is related to the concept of, that is, the matrix eigenvectors (1 ± Su)/2 are or  and therefore each generates an ideal in the Pauli algebra. The same trick works in any, in particular the that are discussed below. These projection operators are also seen in theory where they are examples of pure density matrices.

More generally, the projection operator for spin in the (a, b, c) direction is given by
 * $$\frac{1}{2}\begin{bmatrix}1+c&a-ib\\a+ib&1-c\end{bmatrix}$$

and any non zero column can be taken as the projection operator. While the two columns appear different, one can use a2 + b2 + c2 = 1 to show that they are multiples (possibly zero) of the same spinor.


 * From Tensor:

When changing from one (called a frame) to another by a rotation, the components of a tensor transform by that same rotation. This transformation does not depend on the path taken through the space of frames. However, the space of frames is not (see  and ): there are continuous paths in the space of frames with the same beginning and ending configurations that are not deformable one into the other. It is possible to attach an additional discrete invariant to each frame that incorporates this path dependence, and which turns out (locally) to have values of ±1. A is an object that transforms like a tensor under rotations in the frame, apart from a possible sign that is determined by the value of this discrete invariant.

Succinctly, spinors are elements of the of the rotation group, while tensors are elements of its. Other have tensor representations, and so also tensors that are compatible with the group, but all non-compact classical groups have infinite-dimensional unitary representations as well.


 * From Spinor:

"Quote from Elie Cartan: The Theory of Spinors, Hermann, Paris, 1966: "Spinors...provide a linear representation of the group of rotations in a space with any number $n$ of dimensions, each spinor having $2^\nu$ components where $n = 2\nu+1$ or $2\nu$." The star (*) refers to Cartan 1913."

(Note: $$\nu$$ is the number of an object can have in n dimensions.)

Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra. The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest to work with. After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations. The spinors are the column vectors on which these matrices act. In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices, and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way. As a representation of the spin group, this realization of spinors as (complex) column vectors will either be irreducible if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even.

In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of Pauli spin matrices corresponding to (angular momenta about) the three coordinate axes. These are 2×2 matrices with complex entries, and the two-component complex column vectors on which these matrices act by matrix multiplication are the spinors. In this case, the spin group is isomorphic to the group of 2×2 unitary matrices with determinant one, which naturally sits inside the matrix algebra. This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves, realizing it as a group of rotations among them, but it also acts on the column vectors (that is, the spinors).


 * From Spinor:

In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles. More precisely, it is the fermions of spin-1/2 that are described by spinors, which is true both in the relativistic and non-relativistic theory. The wavefunction of the non-relativistic electron has values in 2 component spinors transforming under three-dimensional infinitesimal rotations. The relativistic for the electron is an equation for 4 component spinors transforming under infinitesimal Lorentz transformations for which a substantially similar theory of spinors exists.

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