Pseudo-determinant explained

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:

| A | _{+} = \lim _{\alpha\to} (| A + \alpha I |)/(\alpha ^{n-\operatorname{rank} ( A )) }

where |A| denotes the usual determinant, I denotes the identity matrix and rank(A) denotes the matrix rank of A.^[2]

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 l{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 | 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.^[3] Thus pseudo-determinants are sometime presented in the outputs of statistical programs in cases where covariance matrices are singular.^[4] In particular, the normalization for a multivariate normal distribution with a covariance matrix that is not necessarily nonsingular can be written as$$\backslash frac\; =\; \backslash frac\backslash ,.$$

See also

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

References

1. Web site: Minka, T.P. . Inferring a Gaussian Distribution . 2001. PDF
2. Book: Florescu, Ionut . Probability and Stochastic Processes . Wiley . 2014 . 529 .
3. http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/viewer.htm#statug_rreg_sect021.htm SAS documentation on "Robust Distance"
4. Bohling, Geoffrey C. (1997) "GSLIB-style programs for discriminant analysis and regionalized classification", Computers & Geosciences, 23 (7), 739 - 761