Matrix theory

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse.

vectors that map to their scalar multiples, and the associated scalars

sum of the elements of the main diagonal of a square matrix

square matrix with non-zero determinant

mathematical operation in linear algebra

subclass of determinant

theorem that a square matrix satisfies its own characteristic equation

binary operation on matrices

theorem in linear algebra that a system of linear equations with n variables has solution(s) iff the rk(A) = rk([A|b]), and that if there are solutions, they form an affine space of dimension n−rk(A)

for a square matrix, the transpose of the cofactor matrix

matrix decomposition

change of coordinates for a vector space

n×n determinant as sum of n minors weighted by cofactor from row and column not in minor

form of a matrix indicating its eigenvalues and their algebraic multiplicities

minimal polynomial of a matrix

theorem of matrix algebra of invariance properties under basis transformations

matrix operation generalizing exponentiation of scalar numbers

representation of a matrix as a product

matrix operation

theorem

mathematical theorem about eigenvalues

mathematical operation

matrix decomposition

representation of invertible matrices as unitaries multiplying a Hermitian operator

matrix with a specific relation to its characteristic polynomial p

theorem

polynomial of the elements of a matrix

matrix decomposition

function that maps matrices to matrices

mathematical operation on invertible matrices

the set of the matrix's eigenvalues

given a matrix A, the unique matrix B such that ABA = A, BAB = B, and that AB and BA are both Hermitian

block diagonal matrix of Jordan blocks

criterion of positive definiteness of a Hermitian matrix

vector satisfying some of the criteria of an eigenvector

specialized notation for multivariable calculus

ring of all matrices of fixed size

algorithms for matrix decomposition

algorithm

theorem

In mathematics, a unipotent element r of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n.

formula for the derivative of the determinant of a matrix

matrix normal form

In mathematics, a -matrix is a complex square matrix with every principal minor is positive. A closely related class is that of P_0-matrices, which are the closure of the class of -matrices, with every principal minor \geq 0.

square matrix such that the determinant of any square submatrix, including the minors, is positive

trace inequality between traces of exponential of symmetric matrices

formula of matrix exponentials

algorithm to multiply matrices

Binary operation, takes two matrices and returns a scalar

formula computing the inverse of the sum of a matrix with the outer product of two vectors

Tool in control theory

characterizes the diagonal of a Hermitian matrix with given eigenvalues

For two suitable matrices, A and B, I+AB and I+BA have the same determinate

Randomized algorithm for verfiying matrix multiplication

a polynomial with square matrices as variables

type of product of matrices

In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant and permanent.

sets of matrices whose products do not depend on the order of multiplication