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:
 * $$|\mathbf{A}|_+ = \lim_{\alpha\to 0} \frac{|\mathbf{A} + \alpha \mathbf{I}|}{\alpha^{n-\operatorname{rank}(\mathbf{A})}}$$

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. $$(ax + b)(cx + d)^{-1}$$ for $$a, b, c, d \in \mathcal{G}(p, q)$$), is defined as $$[f] = \begin{bmatrix}a & b \\c & d \end{bmatrix}$$. By the pseudo-determinant of the Vahlen matrix for the conformal transformation, we mean
 * $$ \operatorname{pdet} \begin{bmatrix}a & b\\ c& d\end{bmatrix} = ad^\dagger - bc^\dagger.$$

If $$\operatorname{pdet}[f] > 0$$, the transformation is sense-preserving (rotation) whereas if the $$\operatorname{pdet}[f] < 0$$, 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 $$|\mathbf{A}|_+$$ may be computed as the product of the non-zero singular values. If all singular values are zero, then the pseudo-determinant is 1.

Supposing $$\operatorname{rank}(A) = k $$, so that k is the number of non-zero singular values, we may write $$A = PP^\dagger$$ 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 $$|A|_+ = \left|P^\dagger P\right|$$, where $$\left|P^\dagger P\right|$$ is the usual determinant in k dimensions. Further, if $$P$$ is written as the block column $$P = \left(\begin{smallmatrix} C \\ D \end{smallmatrix}\right)$$, then it holds, for any heights of the blocks $$C$$ and $$D$$, that $$|A|_+ = \left|C^\dagger C + D^\dagger D\right|$$.

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 $$\frac{1}{\sqrt{(2\pi)^{\operatorname{rank}(\mathbf\Sigma)}|\mathbf\Sigma|_+}} = \frac{1}{\sqrt{|2\pi\mathbf\Sigma|_+}}\,.$$