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]

References

↑ Minka, T.P. (2001). "Inferring a Gaussian Distribution". PDF
↑ SAS documentation on "Robust Distance"
↑ 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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