For A sym. pos. def. and D pos. diagonal, when is AD pos. def?
If this is better placed in the general analysis forum, please tell me. I'm new here.
Expanding the Subject:
- real square symmetric positive definite.
- real square positive diagonal.
Under what conditions (sufficient nice; necessary excellent) is 
also positive definite?
It is easy to show AD has positive eigenvalues, so the "failure" case is when a negative one pops up in
^T)
.
I have not been able to get traction on this. All help appreciated.
Background: The need is to explore convergence of a very large, very nonlinear system of differential equations. My best shot at a Lyapunov function has derivative of the form
![\frac{dE}{dt} = - \vec{f}(\vec{u})^T [AD(u)] \vec{f}(\vec{u})](http://latex.codecogs.com/png.latex?\frac{dE}{dt} = - \vec{f}(\vec{u})^T [AD(u)] \vec{f}(\vec{u}))
.
The system is bounded, so if AD is always positive definite, it will converge. Alas, it starts meandering in experiments exactly when AD becomes indefinite. I'm searching for a fix.
Many thanks for your thoughts.
Re: For A sym. pos. def. and D pos. diagonal, when is AD pos. def?
Quote:
Originally Posted by
Gene 
If this is better placed in the general analysis forum, please tell me. I'm new here.
Expanding the Subject:
- real square symmetric positive definite.
- real square positive diagonal.
Under what conditions (sufficient nice; necessary excellent) is
also positive definite?
It is easy to show AD has positive eigenvalues, so the "failure" case is when a negative one pops up in
.
I have not been able to get traction on this. All help appreciated.
Background: The need is to explore convergence of a very large, very nonlinear system of differential equations. My best shot at a Lyapunov function has derivative of the form
.
The system is bounded, so if AD is always positive definite, it will converge. Alas, it starts meandering in experiments exactly when AD becomes indefinite. I'm searching for a fix.
Many thanks for your thoughts.
Ok, so you know that has positive eigenvalues, and so if it's Hermitian, then you're done. But, both and are Hermitian, so $A,D$ commuting seems to be what you're looking for. Am I missing something?
