Pseudo-determinant

Last updated July 10, 2024

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 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 $a,b,c,d\in {\mathcal {G}}(p,q)$ ), 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.

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 $Σ$ that is not necessarily nonsingular can be written as ${\frac {1}{\sqrt {(2\pi )^{\operatorname {rank} (\mathbf {\Sigma } )}|\mathbf {\Sigma } |_{+}}}}={\frac {1}{\sqrt {|2\pi \mathbf {\Sigma } |_{+}}}}\,.$

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References

↑ Minka, T.P. (2001). "Inferring a Gaussian Distribution". PDF
↑ Florescu, Ionut (2014). Probability and Stochastic Processes. Wiley. p. 529.
↑ 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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[minka-1] Minka, T.P. (2001). "Inferring a Gaussian Distribution". PDF

[2] Florescu, Ionut (2014). Probability and Stochastic Processes. Wiley. p. 529.

[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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