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:

$|\mathbf{A}{|}_{+}=\underset{\alpha \to 0}{\lim }\frac{|\mathbf{A}+\alpha \mathbf{I}|}{{\alpha }^{n-\mathrm{rank}(\mathbf{A})}}$

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 $[f]=::\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$. By the pseudo determinant of the Vahlen matrix for the conformal transformation, we mean

$pdet::\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=a{d}^{†}-b{c}^{†}$

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

I am Chester from Den Haag. I am learning to play the Cello. Other hobbies are Running.

Also visit my website: Hostgator Coupons - dawonls.dothome.co.kr -