Pseudo determinant

Definition

The pseudo-determinant of a square n-by-n matrix A may be defined as:

where |A| denotes the usual determinant, I denotes the identity matrix and rank(A) denotes the rank of A.

Definition of pseudo determinant using Vahlen matrix

The Vahlen matrix of a conformal transformation, the Möbius transformation (i.e. ( a x + b ) ( c x + d ) − 1 for a , b , c , d ∈ G ( p , q ) )) is defined as [ f ] = [ a b c d ] . By the pseudo determinant of the Vahlen matrix for the conformal transformation, we mean

If pdet ⁡ [ f ] > 0 , the transformation is sense-preserving (rotation) whereas if the pdet ⁡ [ f ] < 0 , the transformation is sense-preserving (reflection).

Computation for positive semi-definite case

If A is positive semi-definite, then the singular values and eigenvalues of A coincide. In this case, if the singular value decomposition (SVD) is available, then | A | + may be computed as the product of the non-zero singular values. If all singular values are zero, then the pseudo-determinant is 1.

Application in statistics

If a statistical procedure ordinarily compares distributions in terms of the determinants of variance-covariance matrices then, in the case of singular matrices, this comparison can be undertaken by using a combination of the ranks of the matrices and their pseudo-determinants, with the matrix of higher rank being counted as "largest" and the pseudo-determinants only being used if the ranks are equal. Thus pseudo-determinants are sometime presented in the outputs of statistical programs in cases where covariance matrices are singular.