Adjoint of a matrix

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 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... 

B Complex Conjugate, Transpose, and Hermitian Adjoint of a Matrix... 

Before discussing how to calculate the inverse of a matrix, we will introduce the concept of adjoint of a matrix . The adjoint of a matrix is the matrix of cofactors of the transpose of the matrix. The adjoint of the matrix can be found by using the following steps ... 

In thi.-. case the adjoint matrix is the same as the matrix of cofactors (as A is a symmetric. njlri.x). The inverse of a matrix is obtained by dividing the elements of the adjoint matrix tlie determinant ... 

The algebra of matrices gives rules for (1) equality, (2) addition and subtraction, (3) multiplication, and (4) division as well as (5) an associative and a distributive law. It also includes definitions of (6) a transpose, adjoint and inverse of a matrix. 

Transpose, adjoint, and inverse of a matrix. The inverse has been... 

Before defining the adjoint matrix we must define the transpose of a matrix. This is a matrix of which the columns are the rows, and vice versa, of the original matrix. Symbolically, the transpose of the matrix [ay] is [a Now, the adjoint matrix of a matrix [ay] is defined as follows ... 

Transpose, adjoint, and inverse of a matrix. The inverse has been defined in (4) above and the transpose and adjoint are defined in Appendix A.4-1 and Table 4-1.1. The reader is left to prove (see problem 4.1) that the transpose, adjoint, and inverse of the product of two matrices are given by ... 

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. 

Physicists call A the adjoint of A, a name that is used by mathematicians to refer to an entirely different matrix.) An example is... 

The adjoint of a column matrix is a row matrix containing the complex conjugates of the elements of the column matrix. 

Since the labels a, b, and c are arbitrary, we have shown that the matrix representation of 0 is the adjoint of the matrix representation of 0 since... 

The complex conjugate transpose of a matrix is called the adjoint and is designated with a superscript t. Thus, A+ = AT ... 

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