Matrix trace minimization and zeros

[GoodSpirit]

Hello, :)

I would like to minimize and find the zeros of the function F(S,P)=trace(S-SP’(A+ PSP’)^-1PS) in respect to S and P.

S is symmetric square matrix.
P is a rectangular matrix

Could you help me?
Thank you very much

All the best

GoodSpirit

 

[GoodSpirit]

Hello everybody,

Perhaps I should explain a little bit.

The aim is to minimize an error metric and preferentially drive it to zero.
This should be done as function of S and P, as function of their rank and dimensions in particular.
By the way, the matrix A is symmetric too.

Many thanks

 

[GoodSpirit]

Hello,

Trying to update the equation presentation.

$$F(S,P)=tr(S-S P^T(A+PSP^T)^-1 PS)$$

A is positive definite

I've using matrix derivatives

What do you think?

All the best

GoodSpirit

 

[GoodSpirit]

LateX didn't work here

How to present an equation here?

Thank you

Good Spirit

 

[blue_raver22]

Hello GoodSpirit,

Thank you for your interest in matrix trace minimization and finding zeros. This is a common problem in mathematics and has applications in various fields such as optimization, control theory, and signal processing.

To start, let's define some terms. The trace of a matrix is the sum of its diagonal elements. In this case, we are trying to minimize the trace of a function F(S,P) with respect to two variables, S and P. The function itself involves the matrices S and P, as well as the matrix A and its inverse. The goal is to find values for S and P that make the trace of the function as small as possible.

Finding the zeros of this function means finding the values of S and P that make the function equal to zero. This is also known as solving the equations F(S,P)=0. These zeros can represent critical points or optimal solutions of the function.

To solve this problem, we can use techniques from matrix calculus, such as gradient descent, to iteratively update the values of S and P until we reach a minimum trace or a zero value for the function. Alternatively, we can also use numerical methods like Newton's method or the conjugate gradient method to find the critical points of the function.

In some cases, the function may have multiple local minima or zeros, and it is important to carefully consider the starting values of S and P to ensure we are finding the global minimum or zero. Additionally, the shape and properties of the matrices S, P, and A can also affect the convergence and accuracy of the solution.

I hope this helps you understand the problem better. If you need further assistance, I recommend consulting with a mathematician or a specialist in matrix optimization. Best of luck in your research!