Re: Condition on singular or nearly singular correlation matrices

mzhang33@xxxxxxxxx wrote:
Hi there,

I heard that there is a theorem saying that for a correlation matrix,
if all the off diaganol elements are very small, then it cannot be
nearly singular. The only way to make it nearly singular is to increase
all the off diaganol elements to values close to 1. However I failed to
find the name of the therom. Does anyone know it? Any reference?
Thanks!

Mike


I don't think *all* of the off diagonal elements need to be
made large, merely enough of them, as a corollary of the
Gershgorin circle theorem:

(http://en.wikipedia.org/wiki/Gershgorin_circle_theorem)

Definition: Let A be a complex n x n matrix with engries
a_ij. For i = 1...n, the ith Gershgorin disc of A is the
disc D(a_ii, R_i) centered at a_ii, with radius

R_i = sum(|a_ij|, j != i)

that is, R_i is the sum of the complex magnitudes of the
elements of the ith row if A, excluding the diagonal a_ii.

THEOREM: Every eigenvalue of A belongs to some Gershgorin
disc of A.

Note that the theorem also applies to the transpose of A as well,
so one could just as well take the column sum instead of the row
sum.

Note that in your case (a correlation matrix) each diagonal entry
a_ii is 1, so if the sum of the ith row excluding a_ii is strictly
less than 1, for each i, none of the Gershgorin discs contains 0,
and so 0 cannot be an eigenvalue of the matrix.

There may well be other, more precise, theorems that constrain
the situation more than this one, but this is what came to mind.

Dale
.

Relevant Pages