Re: Applying a condition to a symbolic matrix

• To: mathgroup at smc.vnet.net
• Subject: [mg117628] Re: Applying a condition to a symbolic matrix
• From: Peter Pein <petsie at dordos.net>
• Date: Tue, 29 Mar 2011 06:49:43 -0500 (EST)
• References: <imfaek\$iuj\$1@smc.vnet.net>

```Am 24.03.2011 12:37, schrieb Denis:
> 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

Hi Denis,

the constraints you tell Mathematica with /; regard Patterns. You can
get the region directly.

As it seems easier to visualize the area where the matrix is not
pos.def. I wrote v.H.v <= 0 in the following:

hes = D[Exp[Times @@ (vars = x /@ Range[5])], {vars, 2}];

notposdefregion =
Reduce[vars.hes.vars <= 0 && vars.vars == 10 &&
x[2] x[3] == 5 x[4] x[5], vars[[1 ;; 3]], vars[[4 ;; 5]], Reals];

of course one can try to eliminate any other two variables (if any).

RegionPlot3D[
N[notposdefregion] /.
HoldPattern[a == b] :> Abs[a - b] < 1/20, ##] & @@
Table[{v, -3.2, 3.2}, {v, vars[[1 ;; 3]]}]

well, this needs a while but it seems to work.

Peter

```

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