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 ${\displaystyle [f]={\begin{bmatrix}a&b\\c&d\end{bmatrix}}}$. 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 $A$ 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]

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". 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