# Algebra

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## Algebra Topics

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### Infinitesimal matrix change

Let , , and be square matrices with small, and define(1)where is the identity matrix. Then the inverse of is approximately(2)This can be seen by multiplying(3)(4)(5)(6)Note that if we instead let , and look for an inverse of the form , we obtain(7)(8)(9)(10)In order to eliminate the term, we require . However, then , so so there can be no inverse of this form.The exact inverse of can be found as follows.(11)so(12)Using a general matrix inverse identity then gives(13)

### Matrix power

The power of a matrix for a nonnegative integer is defined as the matrix product of copies of ,A matrix to the zeroth power is defined to be the identity matrix of the same dimensions, . The matrix inverse is commonly denoted , which should not be interpreted to mean .

### Strassen formulas

The usual number of scalar operations (i.e., the total number of additions and multiplications) required to perform matrix multiplication is(1)(i.e., multiplications and additions). However, Strassen (1969) discovered how to multiply two matrices in(2)scalar operations, where is the logarithm to base 2, which is less than for . For a power of two (), the two parts of (2) can be written(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)so (◇) becomes(13)Two matrices can therefore be multiplied(14)(15)with only(16)scalar operations (as it turns out, seven of them are multiplications and 18 are additions). Define the seven products (involving a total of 10 additions) as(17)(18)(19)(20)(21)(22)(23)Then the matrix product is given using the remaining eight additions as(24)(25)(26)(27)(Strassen 1969, Press et al. 1989).Matrix inversion of a matrix to yield can also be done in fewer operations than expected using the formulas(28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38)(Strassen..

### Matrix multiplication

The product of two matrices and is defined as(1)where is summed over for all possible values of and and the notation above uses the Einstein summation convention. The implied summation over repeated indices without the presence of an explicit sum sign is called Einstein summation, and is commonly used in both matrix and tensor analysis. Therefore, in order for matrix multiplication to be defined, the dimensions of the matrices must satisfy(2)where denotes a matrix with rows and columns. Writing out the product explicitly,(3)where(4)(5)(6)(7)(8)(9)(10)(11)(12)Matrix multiplication is associative, as can be seenby taking(13)where Einstein summation is again used. Now, since , , and are scalars, use the associativity of scalar multiplication to write(14)Since this is true for all and , it must be true that(15)That is, matrix multiplication is associative. Equation(13) can therefore be written(16)without ambiguity. Due to associativity,..

### Hermitian part

Every complex matrix can be broken into a Hermitianpart(i.e., is a Hermitian matrix) and an antihermitian part(i.e., is an antihermitian matrix). Here, denotes the adjoint.

### Square root method

The square root method is an algorithm which solves the matrixequation(1)for , with a symmetric matrix and a given vector. Convert to a triangular matrix such that(2)where is the transpose. Then(3)(4)so(5)giving the equations(6)(7)(8)(9)(10)These give(11)(12)(13)(14)(15)giving from . Now solve for in terms of the s and ,(16)(17)(18)which gives(19)(20)(21)Finally, find from the s and ,(22)(23)(24)giving the desired solution,(25)(26)(27)

### Matrix inverse

The inverse of a square matrix , sometimes called a reciprocal matrix, is a matrix such that(1)where is the identity matrix. Courant and Hilbert (1989, p. 10) use the notation to denote the inverse matrix.A square matrix has an inverse iff the determinant (Lipschutz 1991, p. 45). The so-called invertible matrix theorem is major result in linear algebra which associates the existence of a matrix inverse with a number of other equivalent properties. A matrix possessing an inverse is called nonsingular, or invertible.The matrix inverse of a square matrix may be taken in the Wolfram Language using the function Inverse[m].For a matrix(2)the matrix inverse is(3)(4)For a matrix(5)the matrix inverse is(6)A general matrix can be inverted using methods such as the Gauss-Jordan elimination, Gaussian elimination, or LU decomposition.The inverse of a product of matrices and can be expressed in terms of and . Let(7)Then(8)and(9)Therefore,(10)so(11)where..

### Matrix exponential

The power series that defines the exponential map also defines a map between matrices. In particular,(1)(2)(3)converges for any square matrix , where is the identity matrix. The matrix exponential is implemented in the Wolfram Language as MatrixExp[m].The Kronecker sum satisfies the nice property(4)(Horn and Johnson 1994, p. 208).Matrix exponentials are important in the solution of systems of ordinary differential equations (e.g., Bellman 1970).In some cases, it is a simple matter to express the matrix exponential. For example, when is a diagonal matrix, exponentiation can be performed simply by exponentiating each of the diagonal elements. For example, given a diagonal matrix(5)The matrix exponential is given by(6)Since most matrices are diagonalizable,it is easiest to diagonalize the matrix before exponentiating it.When is a nilpotent matrix, the exponential is given by a matrix polynomial because some power of vanishes...

### Matrix equality

Two matrices and are said to be equal iff(1)for all . Therefore,(2)while(3)

### Matrix direct sum

The matrix direct sum of matrices constructs a block diagonal matrix from a set of square matrices, i.e.,(1)(2)

Denote the sum of two matrices and (of the same dimensions) by . The sum is defined by adding entries with the same indicesover all and . For example,Matrix addition is therefore both commutative andassociative.

### Kronecker sum

The Kronecker sum is the matrix sum defined by(1)where and are square matrices of order and , respectively, is the identity matrix of order , and denotes the Kronecker product.For example, the Kronecker sum of two matrices and is given by(2)The Kronecker sum satisfies the nice property(3)where denotes a matrix exponential.

### Antihermitian part

Every complex matrix can be broken into a Hermitian part(i.e., is a Hermitian matrix) and an antihermitian part(i.e., is an antihermitian matrix). Here, denotes the conjugate transpose.

### Natural norm

Let be a vector norm of a vector such thatThen is a matrix norm which is said to be the natural norm induced (or subordinate) to the vector norm . For any natural norm,where is the identity matrix. The natural matrix norms induced by the L1-norm, L2-norm, and L-infty-norm are called the maximum absolute column sum norm, spectral norm, and maximum absolute row sum norm, respectively.

### Maximum absolute row sum norm

The natural norm induced by the L-infty-normis called the maximum absolute row sum norm and is defined byfor a matrix . This matrix norm is implemented as Norm[m, Infinity].

### Positive matrix

A positive matrix is a real or integer matrix for which each matrix element is a positive number, i.e., for all , .Positive matrices are therefore a subset of nonnegativematrices.Note that a positive matrix is not the same as a positivedefinite matrix.

### Gram matrix

Given a set of vectors (points in ), the Gram matrix is the matrix of all possible inner products of , i.e.,where denotes the transpose.The Gram matrix determines the vectors up to isometry.

### Unimodular matrix

A unimodular matrix is a real square matrix with determinant (Born and Wolf 1980, p. 55; Goldstein 1980, p. 149). More generally, a matrix with elements in the polynomial domain of a field is called unimodular if it has an inverse whose elements are also in . A matrix is therefore unimodular iff its determinant is a unit of (MacDuffee 1943, p. 137).The matrix inverse of a unimodular realmatrix is another unimodular matrix.There are an infinite number of unimodular matrices not containing any 0s or . One parametric family is(1)Specific examples of unimodular matrices having small positive integer entries include(2)(Guy 1989, 1994).The th power of a unimodular matrix(3)is given by(4)where(5)and the are Chebyshev polynomials of the second kind,(6)(Born and Wolf 1980, p. 67)...

### Tridiagonal matrix

A square matrix with nonzero elements only on the diagonal and slots horizontally or vertically adjacent the diagonal (i.e., along the subdiagonal and superdiagonal),Computing the determinant of such a matrix requires only (as opposed to ) arithmetic operations (Acton 1990, p. 332). Efficient solution of the matrix equation for , where is a tridiagonal matrix, can be performed in the Wolfram Language using LinearSolve on , represented as a SparseArray.

### Positive definite matrix

An complex matrix is called positive definite if(1)for all nonzero complex vectors , where denotes the conjugate transpose of the vector . In the case of a real matrix , equation (1) reduces to(2)where denotes the transpose. Positive definite matrices are of both theoretical and computational importance in a wide variety of applications. They are used, for example, in optimization algorithms and in the construction of various linear regression models (Johnson 1970).A linear system of equations with a positive definite matrix can be efficiently solved using the so-called Cholesky decomposition. A positive definite matrix has at least one matrix square root. Furthermore, exactly one of its matrix square roots is itself positive definite.A necessary and sufficient condition for a complex matrix to be positive definite is that the Hermitian part(3)where denotes the conjugate transpose, be positive definite. This means that a real matrix..

### Triangular matrix

An upper triangular matrix is defined by(1)Written explicitly,(2)A lower triangular matrix is defined by(3)Written explicitly,(4)

### Permutation matrix

A permutation matrix is a matrix obtained by permuting the rows of an identity matrix according to some permutation of the numbers 1 to . Every row and column therefore contains precisely a single 1 with 0s everywhere else, and every permutation corresponds to a unique permutation matrix. There are therefore permutation matrices of size , where is a factorial.The permutation matrices of order two are given by(1)and of order three are given by(2)A permutation matrix is nonsingular, and the determinant is always . In addition, a permutation matrix satisfies(3)where is a transpose and is the identity matrix.Applied to a matrix , gives with rows interchanged according to the permutation vector , and gives with the columns interchanged according to the given permutation vector.Interpreting the 1s in an permutation matrix as rooks gives an allowable configuration of nonattacking rooks on an chessboard. However, the permutation matrices provide..

### Toeplitz matrix

Given numbers , where , ..., , 0, 1, ..., , a Toeplitz matrix is a matrix which has constant values along negative-sloping diagonals, i.e., a matrix of the formMatrix equations ofthe formcan be solved with operations. Typical problems modelled by Toeplitz matrices include the numerical solution of certain differential and integral equations (regularization of inverse problems), the computation of splines, time series analysis, signal and image processing, Markov chains, and queuing theory (Bini 1995).

### Periodic matrix

A square matrix such that the matrix power for a positive integer is called a periodic matrix. If is the least such integer, then the matrix is said to have period . If , then and is called idempotent.

### Equivalent matrix

Two matrices and are equal to each other, written , if they have the same dimensions and the same elements for , ..., and , ..., .Gradshteyn and Ryzhik (2000) call an matrix "equivalent" to another matrix ifffor and any suitable nonsingular and matrices, respectively.

### Pauli matrices

The Pauli matrices, also called the Pauli spin matrices, are complex matrices that arise in Pauli's treatment of spin in quantum mechanics. They are defined by(1)(2)(3)(Condon and Morse 1929, p. 213; Gasiorowicz 1974, p. 232; Goldstein 1980, p. 156; Liboff 1980, p. 453; Arfken 1985, p. 211; Griffiths 1987, p. 115; Landau and Lifschitz 1991, p. 204; Landau 1996, p. 224).The Pauli matrices are implemented in the Wolfram Language as PauliMatrix[n], where , 2, or 3.The Pauli spin matrices satisfy the identities(4)(5)(6)where is the identity matrix, is the Kronecker delta, is the permutation symbol, the leading is the imaginary unit (not the index ), and Einstein summation is used in (6) to sum over the index (Arfken 1985, p. 211; Griffiths 1987, p. 139; Landau and Lifschitz 1991, pp. 204-205).The Pauli matrices plus the identity matrix form a complete set, so any matrix..

### Elementary matrix

An matrix is an elementary matrix if it differs from the identity by a single elementary row or column operation.

### Submatrix

A submatrix of an matrix (with , ) is a matrix formed by taking a block of the entries of this size from the original matrix.

### Strictly upper triangular matrix

A strictly upper triangular matrix is an upper triangular matrix having 0s along the diagonal as well as the lower portion, i.e., a matrix such that for . Written explicitly,

### Doubly stochastic matrix

A doubly stochastic matrix is a matrix such that andis some field for all and . In other words, both the matrix itself and its transpose are stochastic.The following tables give the number of distinct doubly stochastic matrices (and distinct nonsingular doubly stochastic matrices) over for small .doubly stochastic matrices over 21, 2, 16, 512, ...31, 3, 81, ...41, 4, 256, ...doubly stochastic nonsingular matrices over 21, 2, 6, 192, ...31, 2, 54, ...41, 4, 192, ...Horn (1954) proved that if , where and are complex -vectors, is doubly stochastic, and , , ..., are any complex numbers, then lies in the convex hull of all the points , , where is the set of all permutations of . Sherman (1955) also proved the converse.Birkhoff (1946) proved that any doubly stochastic matrix is in the convex hull of permutation matrices for . There are several proofs and extensions of this result (Dulmage and Halperin 1955, Mendelsohn and Dulmage 1958, Mirsky 1958, Marcus..

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