Adjoint of matrix

A oonjuga complex of matrix A A e txancpoce of matrix A A aer adjoint of matrix A A inverae of matrix A det(. ) as determinant of matrix A Traoe(. ) sum of diagonal efements of matoix A dtf Kronooker delta (equalx 0 if equals 1... 

A = conjugate complex of matrix A A — transpose of matrix A A = = adjoint of matrix A A l inverse of matrix A... 

Adjoint of a matrix The adjoint of an x matrix is the transpose of the matrix when all elements have been replaeed by their eofaetors. 

The matrix y5 = y°y1y2y3[(y5)2 = +1], anticommutes with all the y s. It therefore has the property that ysyu(y5) 1 — — / . Hence, a possible choice for D is y6. Further properties of the matrices A,B,D can be obtained as follows Consider for example the matrix A. Upon taking the hermitian adjoint of Eq. (9-259) and substituting therein Eq. (9-259) again, we obtain... 

The eigenvalues of a hermitian matrix are real. To prove this statement, we take the adjoint of each side of equation (1.47), apply equation (1.10), and note that A = A ... 

The matrix represented in this chapter by A is usually called the adjoint matrix. It is obtained by constructing the matrix which is composed of all of the cofactors of the elements a,j in A and then taking its transpose. With the basic definition of matrix multiplication (Eq. (29)J and some patience, die reader can verify the relation... 

The adjoint of a matrix is constructed using the cofactors defined earlier. The elements atj of the adjoint matrix A are defined as... 

The adjoint of a matrix is the transpose of the matrix which is formed by replacing each clement with its cofactor, A cofactor is the determinant formed by eliminating the row and column in which each element lies and using the... 

Note that the column vectors v, and the adjoint row vectors v J live in different spaces, so it makes no sense, for example, to add v, and vj. However, according to the rule (9.11) of matrix multiplication, it makes perfect sense to multiply an adjoint v row vector (with... 

An important general concept of matrix algebra is the association of each matrix A with a corresponding adjoint matrix A, defined in (9.14) ... 

There are several types of matrix operations that are used in the MCSCF method. The transpose of a matrix A is denoted A and is defined by (A )ij = Xji. The identity (AB) = B A is sometimes useful where AB implies the usual definition of the product of matrices. A vector, specifically a column vector unless otherwise noted, is a special case of a matrix. A matrix-vector product, as in Eq. (5), is a special case of a matrix product. The conjugate of a matrix is written A and is defined by (A )jj = (A,j). The adjoint, written as A is defined by A = (A ) . The inverse of a square matrix, written as A , satisfies the relation A(A = 1 where = du is called the identity or unit matrix. The inverse of a matrix product satisfies the relation (AB) =B" A" . A particular type of matrix is a diagonal matrix D, where D,y = y, and is sometimes written D = diag(dj, d2> ) or as D = diag(d). The unit matrix is an example of a diagonal matrix. 

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