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-   -   Determine if a matrix if positive definite (http://www.physicsforums.com/showthread.php?t=584315)

 onako Mar6-12 08:15 AM

Determine if a matrix if positive definite

Given a symmetric matrix
$$A=\left(\begin{array}{ccccc} \sum a_{1s} & & & & \\ & \ddots & & a_{ij} \\ & & \ddots & & \\ &a_{ij} & & \ddots & \\ & & & & \sum w_{as} \end{array}\right) \in\mathbb{R}^{n\times n},$$
with strictly positive entries a_{ij}, and with the diagonal entries being sum of off-diagonal entries residing
in the corresponding row/column, how to proceed with the proof for A being positive definite,
$$x^TAx>0$$
for some non-zero vector x.

 AlephZero Mar6-12 04:08 PM

Re: Determine if a matrix if positive definite

The http://en.wikipedia.org/wiki/Gershgorin_circle_theorem shows there are no negative eigenvalues, but it doesn't exclude the possibiltiy of zero eigenvalues (i.e. a singular matrix).

In fact the matrix
$$\begin{pmatrix}1 & 1 \\ 1 & 1 \end{pmatrix}$$
is singular, and therefore not positive definite.

 onako Mar7-12 07:51 AM

Re: Determine if a matrix if positive definite

Thanks for providing the example.
I guess the author of the book stating the above positive-definiteness on the given matrix type
somehow misinterpreted it.

 morphism Mar7-12 01:42 PM

Re: Determine if a matrix if positive definite

Quote:
 Quote by onako (Post 3803087) Thanks for providing the example. I guess the author of the book stating the above positive-definiteness on the given matrix type somehow misinterpreted it.
Actually, unless I'm mistaken, the matrix is invertible (hence positive definite) if n>3.

 Jim Kata Mar7-12 02:05 PM

Re: Determine if a matrix if positive definite

This is true because the matrix is diagonally dominant. There is a theorem that says a Hermitian diagonally dominant matrix with real nonnegative diagonal entries is positive semidefinite. A proof of this is found here http://planetmath.org/?op=getobj&from=objects&id=7483

 onako Mar7-12 02:07 PM

Re: Determine if a matrix if positive definite

It can be shown that the inequality holds for n>3, but not in general case, as is observed above.

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