inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their...

general linear group of all invertible matrices. A triangular matrix is invertible precisely when its diagonal entries are invertible (non-zero). Over the real...

defined using the Leibniz formula; such a matrix is invertible if and only if its determinant is invertible in R, generalizing the situation over a field...

matrix A {\displaystyle A}  is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists an invertible matrix...

in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic product, u vT, of...

,} where I is the identity matrix of the same size as A. Consequently, the multiplicative inverse of an invertible matrix can be found by dividing its...

conditions where "positive" is replaced by "nonnegative", "invertible matrix" is replaced by "matrix", and the word "leading" is removed. Positive-definite...

invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the...

identity matrix. Involutory matrices are all square roots of the identity matrix. This is a consequence of the fact that any invertible matrix multiplied...

m\times n} matrix A {\displaystyle A} . A square matrix A {\displaystyle A} is called invertible or non-singular if there exists a matrix B {\displaystyle...

{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 Q∗ is the...

an invertible element is an element that has an inverse. In a ring, an invertible element, also called a unit, is an element that is invertible under...

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...

words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. When A is an invertible matrix there is...

{2}}\end{bmatrix}}.} An invertible matrix A is a generalized permutation matrix if and only if it can be written as a product of an invertible diagonal matrix D and an...

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

Invertible may refer to Invertible element Invertible function Invertible ideal Invertible knot Invertible jet Invertible matrix Invertible module Invertible...

inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function f : Rn → Rn is the Jacobian matrix of the inverse function. That...

=\mathbf {I} .} A matrix that has an inverse is an invertible matrix. Otherwise, it is a singular matrix. A product of matrices is invertible if and only if...

an invertible matrix. Most rotation matrices fit this description, and for them it can be shown that (Q − I)(Q + I)−1 is a skew-symmetric matrix, A....

The 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...

Neumann series can be used for approximate matrix inversion. To approximate the inverse of an invertible matrix A {\displaystyle A} , consider that A − 1...

the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855). To perform row reduction on a matrix, one uses a sequence...

of 2 matrices, and for sets of matrices which contain at most one invertible matrix. Paterson, Michael S. (1970). "Unsolvability in 3 × 3 matrices". Studies...

factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix multiplication and matrix decomposition)...

\dots ,\ x_{n}} are distinct, then V is a square matrix with non-zero determinant, i.e. an invertible matrix. Thus, given V and y, one can find the required...

matrix similarity. The row echelon form is a canonical form, when one considers as equivalent a matrix and its left product by an invertible matrix....

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 matrix X and adj ⁡ ( X ) {\displaystyle \operatorname {adj} (X)} is its adjugate matrix. (The latter equality only holds if A(t) is invertible.) As...

n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix P such that B = P − 1 A P . {\displaystyle B=P^{-1}AP.} Similar...