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.

## Contents

* 1 Definition
* 2 Definition of pseudo-determinant using Vahlen matrix
* 3 Computation for positive semi-definite case
* 4 Application in statistics
* 5 See also
* 6 References

## Definition

The pseudo-determinant of a square n-by-n matrix A may be defined as:

|𝐀|+=limα->0|𝐀+α𝐈|αn−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∈𝒢(p,q)), is defined as [f]=[abcd]. By the pseudo-determinant of the Vahlen matrix for the conformal transformation, we mean

pdet⁡[abcd]=ad†−bc†.

If pdet⁡[f]>0, the transformation is sense-preserving (rotation) whereas if the 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 rank⁡(A)=k, so that k is the number of non-zero singular values, we may write A=PP† 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|+=|P†P|, where |P†P| is the usual determinant in k dimensions. Further, if P is written as the block column P=(CD), then it holds, for any heights of the blocks C and D, that |A|+=|C†C+D†D|.

## 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 Σ that is not necessarily nonsingular can be written as 1(2π)rank⁡(Σ)|Σ|+=1|2πΣ|+.

## 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". http://research.microsoft.com/en-us/um/people/minka/papers/gaussian.html. PDF
2. ↑ Florescu, Ionut (2014). Probability and Stochastic Processes. Wiley. p. 529. ISBN 978-0-470-62455-5. https://books.google.com/books?id=MaXCBwAAQBAJ&pg=PA529.
3. ↑ 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 doi: 10.1016/S0098-3004(97)00050-2

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