Re: Determinant of a particular symmetric matrix



On Fri, 27 Jan 2006, sajesse wrote:

> Hi all,
> My problem is the following.
> I have a symmetric n x n matrix, which contains only positive values.
> Each value on the main diagonal is equal to 1, and other values v are such that 0 < v < 1.
> Numerically, it seems that the determinant of such a matrix
> is always different from 0, so that it always has an inverse (very interesting to me).
> I'd like to know if there exist a proof of this property (if it's true).
> If someone could help me...
> thanks.

The catch is: with idealized real numbers, if you generate a matrix
according to your description but at random (off-diagonal entries
uniformly distributed over (0,1)), the probability of running into a
singular matrix is zero.

Numerically, this probability is so small (can anyone give an estimate?)
that you might grow very old to get such a matrix singular.

Moreover, such matrices are often indefinite (fail to be positive or
negative definite). Run a pseudorandom experiment to see for yourself. So,
an attempt to argue about eigenvalues is not fruitful.

Hope it helps,
ZVK(Slavek).
.