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(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. formula_2 for formula_3), is defined as formula_4. By the pseudo-determinant of the Vahlen matrix for the conformal transformation, we mean 
If formula_6, the transformation is sense-preserving (rotation) whereas if the formula_7, the transformation is sense-preserving (reflection).
Computation for positive semi-definite case.
If formula_8 is positive semi-definite, then the singular values and eigenvalues of formula_8 coincide. In this case, if the singular value decomposition (SVD) is available, then formula_10 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 formula_11, so that "k" is the number of non-zero singular values, we may write formula_12 where formula_13 is some "n"-by-"k" matrix and the dagger is the conjugate transpose. The singular values of formula_8 are the squares of the singular values of formula_13 and thus we have formula_16, where formula_17 is the usual determinant in "k" dimensions. Further, if formula_13 is written as the block column formula_19, then it holds, for any heights of the blocks formula_20 and formula_20, that formula_22.
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.