Matrix adjugate

Adjugate Matrix of a Matrix Let Ay denote the cofactor of the element Oy in the determinant of the matrix A. The matrix B where B = (Ay) is called the adjugate matrix of A written adj A = B. The elements by are calculated by taking the matrix A, deleting the ith row and Jth. column, and calculating the determinant of the remaining matrix times (—1) Then A" = adj A/lAl. This definition may be used to calculate A"h However, it is very laborious and the inversion is usually accomplished by numerical techniques shown under Numerical Analysis and Approximate Methods. ... 

Mathematics texts use the term transpose conjugate for this matrix and apply the term adjoint to the adjugate matrix defined in equation (1.28). 

If the rows and columns of the cofactor matrix are transposed a matrix AJl, called adjugate to A is obtained, such that... 

For the calculation of cofactors we use algorithms based on work by Lowdin [16], and Prosser and Hagstrom [25,26]. An overview of the theory of determinants, cofactors, adjugates and compound matrices can be found in a book by Aitken [17]. The symmetry and possible orthogonality in the orbital spaces give rise to a block-structure in the overlap matrices. This structure is exploited [22,27] to minimise the size of the matrices in the L-d-R decomposition, described below, an n3 process for each matrix. 

Now the adjugate matrix of S, which contains the first order cofactors can be calculated using [17] and Eq. (38)... 

Using the fact that the inverse of a diagonal matrix is a diagonal matrix with inverse elements, and that the adjugate of a matrix is directly related to its inverse... 

The th order compound matrix B(k) is a matrix with the ktb order minors (A111 order sub-determinants) as elements. So the kth order adjugate can be expressed in terms of the klb order compound matrix of the first order adjugate. 

Now the first order adjugate of the matrix d contains only 1 nonzero element, simplifying the application of Eq. (37). The Jacobi ratio theorem cannot be used straightaway for the higher order cofactors, when the overlap matrix is singular, since its determinant is zero. However, we can make use of the fact that determinants and thus cofactors, which are sub-determinants, are linear in their elements. If we change a matrix element of S as [ 13,28]... 

I.e. the matrix formed from by replacing every element by its cofactor. This is sometimes called the adjugate or adjugate compound matrix. 

The relation R, - = simply implies that the matrix and its transpose are related by a 180° rotation of the matrix about a line through its diagonal. The adjugate is the matrix of the (n — 1) X ( — 1) cofactors ... 

The A matrix is known as the adjugate (or adjoint) matrix of m which may be written adj m. Thus... 

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