invertible matrix (non-singular, non-degenarate or regular) is a square matrix that has an inverse. In other words, if some other matrix is multiplied...

and in particular linear algebra, the Moore–Penrose inverse ⁠ A + {\displaystyle A^{+}} ⁠ of a matrix ⁠ A {\displaystyle A} ⁠, often called the pseudoinverse...

determinant, and the multiplicative inverse of the derivative is replaced by the inverse of the Jacobian matrix. The Jacobian determinant is fundamentally...

The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of matrices...

unknowns are different so that matrix F {\displaystyle F} is not square. However, even a square matrix can have no inverse: matrix F {\displaystyle F} can be...

to the identity matrix I {\displaystyle \mathbf {I} } , the right-hand n × n {\displaystyle n\times n} block is then the inverse matrix A − 1 {\displaystyle...

entries), an invertible matrix is a matrix that has an inverse that is also an integer matrix. Such a matrix is called a unimodular matrix for distinguishing...

In statistics, the inverse matrix gamma distribution is a generalization of the inverse gamma distribution to positive-definite matrices. It is a more...

algebra, the Woodbury matrix identity – named after Max A. Woodbury – says that the inverse of a rank-k correction of some matrix can be computed by doing...

prior for the covariance matrix of a multivariate normal distribution. We say X {\displaystyle \mathbf {X} } follows an inverse Wishart distribution, denoted...

where A−1 is the inverse matrix of A. If A has no inverse, solutions—if any—can be found using its generalized inverse. Matrices and matrix multiplication...

{\displaystyle n\times n} identity matrix. The matrix Ω {\displaystyle \Omega } has determinant + 1 {\displaystyle +1} and its inverse is Ω − 1 = Ω T = − Ω {\displaystyle...

multiplicative inverse. For example, a matrix such that all entries of a row (or a column) are 0 does not have an inverse. If it exists, the inverse of a matrix A...

computes the inverse of a "rank-1 update" to a matrix whose inverse has previously been computed. That is, given an invertible matrix A {\displaystyle...

Q^{\mathrm {T} }=Q^{-1},} where Q−1 is the inverse of Q. An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q∗), where...

left inverse of a square matrix is also a right-inverse (see Invertible matrix theorem). For other proofs, see below. Let A be an n × n matrix with entries...

is unique and is called the inverse matrix of A {\displaystyle A} , denoted A − 1 {\displaystyle A^{-1}} . A square matrix A {\displaystyle A} that is...

generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices...

inverse function. The inverse function is also differentiable, and the inverse function rule expresses its derivative as the multiplicative inverse of...

may be satisfactory. The inverse iteration algorithm requires solving a linear system or calculation of the inverse matrix. For non-structured matrices...

Inverse element Inverse function, a function that "reverses" another function Generalized inverse, a matrix that has some properties of the inverse matrix...

In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix A n × n {\displaystyle {\mathbf...

In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U−1 equals its conjugate transpose U*, that is, if U ∗ U = U...

the formula for the inverse involving the adjugate matrix mentioned below. In this event, the determinant of the inverse matrix is given by det ( A −...

matrix-related notions is about properties of products or inverses of the given matrix. The matrix product of a m-by-n matrix A and a n-by-k matrix B...

transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix. The notation A−T is sometimes...

algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors...

post-multiplying an n-column matrix M, forming MP, permutes the columns of M. Every permutation matrix P is orthogonal, with its inverse equal to its transpose:...

The inverse of this matrix is itself: T i , j − 1 = T i , j . {\displaystyle T_{i,j}^{-1}=T_{i,j}.} Since the determinant of the identity matrix is unity...

formulas for the inverse matrix V − 1 {\displaystyle V^{-1}} . In particular, Lagrange interpolation shows that the columns of the inverse matrix V − 1 = [ 1...