       Applying a condition to a symbolic matrix

• To: mathgroup at smc.vnet.net
• Subject: [mg117597] Applying a condition to a symbolic matrix
• From: Denis <denisvasilescu at gmail.com>
• Date: Thu, 24 Mar 2011 06:27:50 -0500 (EST)

```Hello everyone,

I wish to know how to apply a condition to a symbolic matrix for the
purposes of checking if it is positive definite using PositiveDefiniteQ. A
matrix composed of symbolic expressions may not be positive definite for all
values, but it may be positive definite on a certain domain.

The matrix I wish to check is the Hessian of the function:

f[x1_, x2_, x3_, x4_, x5_] := e^(x1 x2 x3 x4 x5);

which I can get using

ObtainHessian[f_, x_List?VectorQ] := D[f, {x, 2}];
HessianF = ObtainHessian[f[x1, x2, x3, x4, x5], {x1, x2, x3, x4, x5}];

I wish to consider the function and it's Hessian within these constraints:

x1^2 + x2^2 + x3^2 + x4^2 + x5^2 - 10 == 0;
x2 x3 - 5 x4 x5 == 0;

I know of one possible method of doing this and one work-around. The
work-around is to use a constrained minimization function such as
FindMinValue for {x1,x2,x3,x4,x5}.HessianF.{x1,x2,x3,x4,x5} subject to the
constraints and a starting point, and see if the minimum value is positive.

The possible method would be using "/;conditions" to apply the constraints,
but I am not sure where to stick it because it is intended for delayed
evaluations.

Would anyone know how to condition a symbolic matrix? There may be
particular work-arounds for the specific case of determining if it is
positive definite, but it would be lovely to find a solution that is
generally applicable.

Many thanks,
Denis

```

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