# Trace of the inverse of matrix products

• A
• nikozm
In summary, the conversation discusses the condition of matrix A and matrix B having complex-valued zero-mean Gaussian entries that are mutually independent. The question is then posed about proving that the trace of the inverse of (A*B)^{H}*(A*B) is always lower than the trace of the inverse of (A)^{H}*(A). The conversation also mentions the use of the Hermitian transpose and matrix inverse operators. The question is clarified to ask about the truth of this result for any pair of random matrices A and B that have independent complex-valued Gaussian distributions. Any suggestions or ideas on this topic are welcome.
nikozm
Hello,

I am puzzled about the following condition. Assume a matrix A with complex-valued zero-mean Gaussian entries and a matrix B with complex-valued zero-mean Gaussian entries too (which are mutually independent of the entries of matrix A).

Then, how can we prove that Trace{[(A*B)^{H}*(A*B)]^{-1}} is always lower that Trace{[(A)^{H}*(A)]^{-1}} ?

The superscripts {H} and {-1} denote the Hermitian transpose and matrix inverse operator, respectively.

Thank you very much in advance.

What is the definition of *always* here? I could pick any matrices A and B, and they would have some probability of being sampled by the gaussians you describe. So is this result intended to just be true for any pair of matrices A and B?

Yes. I would like to know if (and how) is this result true for generally random matrices A and B (where their elements are particularly independent complex-valued Gaussian distributed).

Any suggestion could be useful. Thanks in advance.

## 1. What is the trace of the inverse of matrix products?

The trace of the inverse of matrix products is a mathematical concept that represents the sum of the diagonal elements of a matrix. It is a measure of the size and shape of a matrix, and can be used to determine properties such as the determinant and eigenvalues of the matrix.

## 2. How is the trace of the inverse of matrix products calculated?

The trace of the inverse of matrix products can be calculated by first finding the inverse of the matrix, then taking the trace of the resulting matrix. This involves finding the reciprocal of each element in the original matrix, and then taking the sum of the diagonal elements of the resulting matrix.

## 3. What is the significance of the trace of the inverse of matrix products?

The trace of the inverse of matrix products has many applications in mathematics and science. It can be used to solve systems of linear equations, calculate the determinant and eigenvalues of a matrix, and determine the stability of a dynamical system. It is also used in fields such as physics, engineering, and computer science.

## 4. Can the trace of the inverse of matrix products be negative?

Yes, the trace of the inverse of matrix products can be negative. This can occur when the matrix has a negative determinant, which means that the matrix is not invertible. In this case, the trace of the inverse of matrix products will be negative and the matrix will not have a well-defined inverse.

## 5. How is the trace of the inverse of matrix products related to other matrix operations?

The trace of the inverse of matrix products is related to other matrix operations such as matrix multiplication, addition, and subtraction. It can also be used to determine the rank and nullity of a matrix, and to calculate the condition number of a matrix. Additionally, the trace of the inverse of matrix products is used in the proof of the Cayley-Hamilton theorem, which states that every square matrix satisfies its own characteristic equation.

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