pseudo-determinant

In linear algebra and statistics, the pseudo-determinantPDF
/ref> 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:
:$, \mathbf, _+ = \lim_ \frac$

where , A, denotes the usual determinant, I denotes the identity matrix and rank(A) denotes the matrix rank of A.

Definition of pseudo-determinant using Vahlen matrix

The Vahlen matrix of a conformal transformation, the Möbius transformation (i.e. $(ax + b)(cx + d)^$ for $a, b, c, d \in \mathcal(p, q)$), is defined as = \begina & b \\c & d \end. By the pseudo-determinant of the Vahlen matrix for the conformal transformation, we mean
: $\operatorname \begina & b\\ c& d\end = ad^\dagger - bc^\dagger.$

If \operatorname > 0, the transformation is sense-preserving (rotation) whereas if the \operatorname < 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 $, \mathbf, _+$ 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 $\operatorname(A) = k$, 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\$, where $\left, P^\dagger P\$ is the usual determinant in ''k'' dimensions. Further, if $P$ is written as the block column $P = \left(\begin C \\ D \end\right)$, then it holds, for any heights of the blocks $C$ and $D$, that $, A, _+ = \left, C^\dagger C + D^\dagger 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. Thus pseudo-determinants are sometime presented in the outputs of statistical programs in cases where covariance matrices are singular.Bohling, Geoffrey C. (1997) "GSLIB-style programs for discriminant analysis and regionalized classification", ''Computers & Geosciences'', 23 (7), 739–761 In particular, the normalization for a multivariate normal distribution with a covariance matrix that is not necessarily nonsingular can be written as
$\frac = \frac{\sqrt{, 2\pi\mathbf\Sigma, _+\,.$

See also

* Matrix determinant
* Moore–Penrose pseudoinverse, which can also be obtained in terms of the non-zero singular values.

References

Covariance and correlation
Matrices (mathematics)