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:

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

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

Definition of pseudo-determinant using Vahlen matrix

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

$\displaystyle{ \operatorname{pdet} \begin{bmatrix}a & b\\ c& d\end{bmatrix} = ad^\dagger - bc^\dagger. }$

If $\displaystyle{ \operatorname{pdet}[f] \gt 0 }$, the transformation is sense-preserving (rotation) whereas if the $\displaystyle{ \operatorname{pdet}[f] \lt 0 }$, the transformation is sense-preserving (reflection).

Computation for positive semi-definite case

If $\displaystyle{ A }$ is positive semi-definite, then the singular values and eigenvalues of $\displaystyle{ A }$ coincide. In this case, if the singular value decomposition (SVD) is available, then $\displaystyle{ |\mathbf{A}|_+ }$ 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 $\displaystyle{ \operatorname{rank}(A) = k }$, so that k is the number of non-zero singular values, we may write $\displaystyle{ A = PP^\dagger }$ where $\displaystyle{ P }$ is some n-by-k matrix and the dagger is the conjugate transpose. The singular values of $\displaystyle{ A }$ are the squares of the singular values of $\displaystyle{ P }$ and thus we have $\displaystyle{ |A|_+ = \left|P^\dagger P\right| }$, where $\displaystyle{ \left|P^\dagger P\right| }$ is the usual determinant in k dimensions. Further, if $\displaystyle{ P }$ is written as the block column $\displaystyle{ P = \left(\begin{smallmatrix} C \\ D \end{smallmatrix}\right) }$, then it holds, for any heights of the blocks $\displaystyle{ C }$ and $\displaystyle{ D }$, that $\displaystyle{ |A|_+ = \left|C^\dagger C + D^\dagger D\right| }$.

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.