       Re: Help with minimization of Eigenvalues

• To: mathgroup at smc.vnet.net
• Subject: [mg87779] Re: Help with minimization of Eigenvalues
• From: "Jean-Marc Gulliet" <jeanmarc.gulliet at gmail.com>
• Date: Wed, 16 Apr 2008 22:32:30 -0400 (EDT)
• References: <fu4ll6\$sfq\$1@smc.vnet.net> <480602EA.8010102@gmail.com>

```Chimico wrote:

> I am trying to minimize a particular eigenvalue of a generalized
> eigenvalue problem that depends on a few parameters. I have the
> following problem:
>
> Suppose we have two matrices s and m like these (the real matrices are
> much more complicated)
>
> m = {{a, b}, {b, -a + 1}}
>
> s = {{1, 0}, {0, 3a}}
>
> If I define the function Eig
>
> Eig[a_, b_] := Eigenvalues[{m, s}]//First
>
> And use it with the FindMinimum function it does not work.
> Infact just calling
>
> Eig[1,2] gives the error
>
> Eigenvalues::gfargs:
>    Generalized Eigenvalues
>      arguments accept only matrices with machine real
>       and complex elements. More...
>
> Even inserting Evaluate[] into the Eigenvalues function does not work.
> What am I doing wrong?

Your function Eig[a, b] does *not* use or pass its arguments to the
matrices m and s. (In other words, the values a and b on the LHS are not
used on the RHS: the matrices still hold the symbols a and b.) The
following is one possible way to modify the value of a and b within the
matrices m and s.

In:= m = {{a, b}, {b, -a + 1}};
s = {{1, 0}, {0, 3 a}};
Eig[p_, q_] := Eigenvalues[N@{m, s} /. a -> p /. b -> q] // First
Eig[1, 2]

Out= 1.75831

Moreover, though I am not sure to have fully understood what you tried
to achieved, you might be interested in using the function
CharacteristicPolynomial[] to compute the eigenvalues in a symbolic form
and use these results to solve or minimize them.

In:= CharacteristicPolynomial[{m, s}, x]

Out= a - a^2 - b^2 - x + a x - 3 a^2 x + 3 a x^2

In:= Reduce[CharacteristicPolynomial[{m, s}, x] == 0]

Out= (x == -(1/3) &&
a == 1/3 (-1 + 3 b^2)) || (1 + 3 x !=
0 && (a == (
1 + x + 3 x^2 -
Sqrt[-4 (b^2 + x) (1 + 3 x) + (-1 - x - 3 x^2)^2])/(
2 (1 + 3 x)) ||
a == (1 + x + 3 x^2 +
Sqrt[-4 (b^2 + x) (1 + 3 x) + (-1 - x - 3 x^2)^2])/(
2 (1 + 3 x))))

Regards
-- Jean-Marc

```

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