# Pseudo-determinant

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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:

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

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

## 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]

## 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