algebra, an invertible matrix is a square matrix which has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result...

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

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

,} 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...

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...

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...

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...

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

A matrix is positive semi-definite if it satisfies similar equivalent conditions where "positive" is replaced by "nonnegative", "invertible matrix" is...

{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...

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...

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...

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....

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

\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...

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...

congruent if there exists an invertible matrix P over the same field such that PTAP = B where "T" denotes the matrix transpose. Matrix congruence is an equivalence...

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...

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...

=\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...

diagonal matrix is invertible if and only if each of its main-diagonal blocks are invertible, and in this case its inverse is another block diagonal matrix given...

of a Lie algebra, when the Lie algebra is gln Invertible matrix (this usage is rare) QS Regular Matrix, a quadraphonic sound system developed by Sansui...

invertible matrices. In fact, this map is surjective which means that every invertible matrix can be written as the exponential of some other matrix (for...

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...

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

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...

all invertible n × n {\displaystyle n\times n} matrices under the matrix multiplication operation. In particular, the identity matrix is invertible. It...

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...

the left by an n × n invertible matrix S, and the symmetric square matrix A is transformed into another symmetric square matrix B of the same size according...