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(A) denotes the matrix rank of A.

Definition of pseudo-determinant using Vahlen matrix

The Vahlen matrix of a conformal transformation, the Möbius transformation (i.e. for ), is defined as. By the pseudo-determinant of the Vahlen matrix for the conformal transformation, we mean If, the transformation is sense-preserving (rotation) whereas if the , 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 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 P is some n-by-k matrix and the dagger is the conjugate transpose. The singular values of A are the squares of the singular values of P and thus we have, where is the usual determinant in k dimensions. Further, if P is written as the block column, then it holds, for any heights of the blocks C and D, that.

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. In particular, the normalization for a multivariate normal distribution with a covariance matrix Σ that is not necessarily nonsingular can be written as