# Matrix trace minimization and zeros

• GoodSpirit
In summary, The conversation involves a request for help in minimizing and finding the zeros of a function F(S,P) in respect to symmetric square matrix S and rectangular matrix P. The function involves a trace of matrices and is dependent on the rank and dimensions of S and P. The matrix A is also mentioned, being symmetric and positive definite. The use of matrix derivatives and difficulties with presenting equations using LateX are also mentioned.
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

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

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

LateX didn't work here

How to present an equation here?

Thank you

Good Spirit

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!

## 1. What is matrix trace minimization?

Matrix trace minimization is a mathematical optimization technique that aims to find the smallest possible value of the sum of the diagonal elements of a matrix. This is achieved by manipulating the other elements of the matrix while keeping the diagonal elements fixed.

## 2. Why is matrix trace minimization important?

Matrix trace minimization has various applications in fields such as machine learning, control theory, and signal processing. It can be used to solve problems such as system identification, state estimation, and parameter estimation.

## 3. How is matrix trace minimization related to zeros of a matrix?

The zeros of a matrix are the values that make the determinant of the matrix equal to zero. In matrix trace minimization, finding the zeros of a matrix can help in identifying the optimal solution where the trace of the matrix is minimized.

## 4. Can matrix trace minimization be used for non-square matrices?

No, matrix trace minimization is only applicable to square matrices, as the trace is defined as the sum of the diagonal elements, which only exists for square matrices.

## 5. What are some techniques for performing matrix trace minimization?

Some common techniques for matrix trace minimization include gradient descent, Newton's method, and the alternating direction method of multipliers (ADMM). These techniques involve iteratively updating the elements of the matrix until the desired minimum trace value is achieved.

• Linear and Abstract Algebra
Replies
2
Views
1K
• Linear and Abstract Algebra
Replies
6
Views
1K
• Linear and Abstract Algebra
Replies
1
Views
771
• Linear and Abstract Algebra
Replies
5
Views
5K
• Linear and Abstract Algebra
Replies
1
Views
1K
• Linear and Abstract Algebra
Replies
1
Views
1K
• Linear and Abstract Algebra
Replies
2
Views
2K
• Linear and Abstract Algebra
Replies
8
Views
2K
• Linear and Abstract Algebra
Replies
2
Views
2K
• Linear and Abstract Algebra
Replies
9
Views
2K