Pseudo-determinant

In linear algebra and statistics, the pseudo-determinant^[1] 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 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.

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.^[2] Thus pseudo-determinants are sometime presented in the outputs of statistical programs in cases where covariance matrices are singular.^[3]

See also

• Matrix determinant
• Moore–Penrose pseudoinverse, which can also be obtained in terms of the non-zero singular values.

References

1. Minka, T.P. (2001). "Inferring a Gaussian Distribution". PDF
2. SAS documentation on "Robust Distance"
3. Bohling, Geoffrey C. (1997) "GSLIB-style programs for discriminant analysis and regionalized classification", Computers & Geosciences, 23 (7), 739–761 doi: 10.1016/S0098-3004(97)00050-2