Algorithmic Lovász local lemma

In linear algebra and statistics, the pseudo-determinant 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

$pdet::{\begin{bmatrix}a&b\\c&d\end{bmatrix}}=ad^{\dagger }-bc^{\dagger }$

If $pdet[f]>0$, the transformation is sense-preserving (rotation) whereas if the $pdet[f]<0$, 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. Thus pseudo-determinants are sometime presented in the outputs of statistical programs in cases where covariance matrices are singular.

See also

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

References

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