the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. It is occasionally known as adjunct matrix, or...

similarly, and the matrix of the Hermitian adjoint is given by the conjugate transpose matrix if the bases are orthonormal. Adjugate matrix, the transpose...

result. For example, one can calculate the inverse of a matrix by computing its adjugate matrix: A − 1 = adj ⁡ ( A ) / det ( A ) . {\displaystyle {\mathbf...

formula for the inverse involving the adjugate matrix mentioned below. In this event, the determinant of the inverse matrix is given by det ( A − 1 ) = 1 det...

conjugate transpose "adjoint" matrix A H {\displaystyle \mathbf {A} ^{\mathrm {H} }} should not be confused with the adjugate, adj ⁡ ( A ) {\displaystyle...

matrix that is its own inverse (i.e., a matrix A such that A = A−1 and consequently A2 = I) is called an involutory matrix. The adjugate of a matrix A...

determinant of an integer matrix M {\displaystyle M} is itself an integer, and the adjugate matrix of an integer matrix is also integer matrix. Thus, the determinant...

)}}\mathbf {C} ^{\mathsf {T}}.} The transpose of the cofactor matrix is called the adjugate matrix (also called the classical adjoint) of A. The above formula...

In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. If...

multiplicity ⌊ n / 2 ⌋ {\displaystyle \lfloor n/2\rfloor } ). The adjugate matrix of Jn is: adj ⁡ ( J n ) = sgn ⁡ ( π n ) J n . {\displaystyle \operatorname...

two vectors u and v. The theorem can also be stated in terms of the adjugate matrix of A: det ( A + u v T ) = det ( A ) + v T a d j ( A ) u , {\displaystyle...

matrix. Recall that its r th higher adjugate matrix adjr (A) is the ( n r ) × ( n r ) {\textstyle {\binom {n}{r}}\!\times \!{\binom {n}{r}}} matrix whose...

matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries...

below use the notion of the adjugate matrix adj(M) of an n × n matrix M, the transpose of its cofactor matrix. This is a matrix whose coefficients are given...

adj(A) denotes the adjugate matrix, det(A) is the determinant, and I is the identity matrix. If det(A) is nonzero, then the inverse matrix of A is A − 1 =...

Adjunction space in topology Conjugate transpose of a matrix in linear algebra Adjugate matrix, related to its inverse Adjoint equation The upper and...

invertible in R. In that case, A−1 can be given explicitly in terms of the adjugate matrix. For elements x and y in a ring R, if 1 − x y {\displaystyle 1-xy}...

where Madj is adjugate matrix given by the standard formula - its (i,j)-th element is the column-determinant of the (n − 1) × (n − 1) matrix that results...

differential of the determinant at an arbitrary square matrix, in terms of the trace and the adjugate of the matrix. From this (or from the connection between the...

an) is the product a1⋯an. The adjugate of a diagonal matrix is again diagonal. Where all matrices are square, A matrix is diagonal if and only if it is...

matrices is regular and the formula for the inverse in terms of the adjugate matrix shows that inversion is regular as well on matrices with determinant...

First, we can directly compute the matrix product M M ′ {\displaystyle MM'} (using simple properties of the adjugate matrix, or alternatively using the formula...

in the lower half plane as well. The analyticity follows from the adjugate matrix formula (since both the entries of Q ( t ) {\displaystyle Q(t)} and...

characteristic polynomial, and a simple transformation allows to find the adjugate matrix in NC. Rearranging the computations into an efficient form leads to...

and dyadic products can all be expressed in matrix form. Dyadic expressions may closely resemble the matrix equivalents. The dot product of a dyadic with...

case this is an elementary fact because the inverse of a matrix is given as the adjugate matrix divided by its determinant). The method of proof here can...

}{2}}(I-A)}} . Adjugate matrix Holomorphic functional calculus Resolvent formalism / Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis...

that for very small problems it is beneficial to replace the matrix inverse with the adjugate, which will yield the same iteration because it is equal to...

abbreviation for adjective, a part of speech AdJ, software Adjugate (or classical adjoint) of a matrix in mathematics Adjukru language, ISO-639-3 code IATA...

{k}}\right)=0.} Notice that the adjugate of a matrix is still a matrix, so the scalar determinant condition has now been replaced by a matrix condition. This would...