Determine if a matrix if positive definite

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Discussion Overview

The discussion revolves around determining whether a given symmetric matrix is positive definite. The matrix in question has strictly positive entries and specific properties related to its diagonal and off-diagonal entries. Participants explore various approaches and theorems related to positive definiteness, including conditions under which the matrix may or may not be positive definite.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a symmetric matrix with strictly positive entries and asks how to prove it is positive definite, specifically that \(x^TAx > 0\) for some non-zero vector \(x\).
  • Another participant references the Gershgorin circle theorem, noting that while it indicates no negative eigenvalues, it does not rule out the possibility of zero eigenvalues, citing the example of a singular matrix.
  • One participant expresses doubt about the positive definiteness claim made in a book, suggesting a misinterpretation, and proposes that the matrix is invertible (and thus positive definite) if \(n > 3\).
  • A different participant asserts that the matrix is diagonally dominant and cites a theorem stating that a Hermitian diagonally dominant matrix with real nonnegative diagonal entries is positive semidefinite.
  • Another participant agrees that the inequality holds for \(n > 3\) but emphasizes that this does not apply to the general case.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which the matrix is positive definite. While some suggest that it may be positive definite for \(n > 3\), others point out that the general case remains unresolved, particularly in light of examples of singular matrices.

Contextual Notes

There are limitations regarding the assumptions made about the matrix's properties, particularly concerning the implications of diagonal dominance and the conditions under which positive definiteness can be established. The discussion highlights the need for careful consideration of matrix size and structure.

onako
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Given a symmetric matrix
A=\left(\begin{array}{ccccc}<br /> \sum a_{1s} &amp; &amp; &amp; &amp; \\<br /> &amp; \ddots &amp; &amp; a_{ij} \\<br /> &amp; &amp; \ddots &amp; &amp; \\<br /> &amp;a_{ij} &amp; &amp; \ddots &amp; \\<br /> &amp; &amp; &amp; &amp; \sum w_{as}<br /> \end{array}\right) \in\mathbb{R}^{n\times n},<br />
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,
<br /> x^TAx&gt;0<br />
for some non-zero vector x.
 
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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.
 
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.
 
onako said:
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.
 
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
 
It can be shown that the inequality holds for n>3, but not in general case, as is observed above.
 

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