Pseudo-determinant
In linear algebra and statistics, the pseudo-determinant is the product of all non-zero eigenvalues of a square matrix. It coincides with the regular determinant when the matrix is non-singular.
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 denotes the matrix rank of A.
Definition of pseudo-determinant using Vahlen matrix
The Vahlen matrix of a conformal transformation, the Möbius transformation, is defined as. By the pseudo-determinant of the Vahlen matrix for the conformal transformation, we meanIf, the transformation is sense-preserving whereas if the, the transformation is sense-preserving.
Computation for positive semi-definite case
If is positive semi-definite, then the singular values and eigenvalues of coincide. In this case, if the singular value decomposition is available, then may be computed as the product of the non-zero singular values. If all singular values are zero, then the pseudo-determinant is 1.Supposing, so that k is the number of non-zero singular values, we may write where is some n-by-k matrix and the dagger is the conjugate transpose. The singular values of are the squares of the singular values of and thus we have, where is the usual determinant in k dimensions. Further, if is written as the block column, then it holds, for any heights of the blocks and, that.