JuliaLang/julia — comment on commit 3678d588e7babeb3dfe76ce3cc96e478a0a2e407
File: j/linalg.j line 277
By ViralBShah:

The following proof was given by John Gilbert. The challenge is how to make a useful algorithm - what tolerances to use - how many trials, etc. It would also be nice to have an analytical result that gives confidence levels. Also, an interesting question is whether this is a useful definition of "How symmetric a matrix is?" when a matrix is not exactly symmetric.

Let $A$ be any $n \times n$ matrix. Let $x$ and $y$ be random vectors
of size $n$ drawn from a normal distribution. The measure of the set of
$x$ and $y$ (as a subset of $\Re^{2n}$) such that $x^* (A-A^*) y = 0$ is
zero.

The biquadratic form $x^* (A-A^*) y$ is a polynomial $p([x; y])$ of degree $2n$.
The $2n$ variables in $p$ are the entries in $x$ and $y$, while the entries
of $A-A^*$ form the co-efficients of the polynomial. For a non--zero
polynomial $p$, the zeros of such a polynomial form a set of measure zero.

-viral

On Jan 3, 2012, at 8:10 PM, Stefan Karpinski wrote:

> ^ *probably.*
>
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> Reply to this email directly or view it on GitHub:
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