# Effect on eigenvalues of multiplying by a diagonal matrix

• Raito
In summary, the conversation discusses the need to compute the eigenvalues of a matrix in order to solve an optimization problem for a MIMO linear precoder. The matrix A is known and the matrix D contains the variables that need to be optimized. Ideas for how to proceed include diagonalizing the matrix B and transforming it to another basis using SD. This may make it easier to compute the eigenvalues.
Raito
Hi,

While trying to solve an optimization problem for a MIMO linear precoder, I have encountered the need to compute the eigenvalues of a matrix $D^{H}A^{H}AD$ where the matrix $A$ is known and the matrix $D$ is a diagonal matrix whose entries contain the variables that need to be optimized (those variables can be assumed to be real without loss of generality).
At first sight, I thought it would be easy but I'm finding myself stuck since any of the ideas I had in mind to do that have been useless.
Any help or idea on how to proceed will be much appreciated.

call A^HA = B

then assuming you can diagonalise B by S then you get something like
$$(D^HS^H)(SBS^H)(SD)$$

then SD is effectively transforming the diagonalised matrix B to another basis. this may make it easier to see how the eigenvalues transform

## 1. How does multiplying a matrix by a diagonal matrix affect its eigenvalues?

Multiplying a matrix by a diagonal matrix does not change the number of eigenvalues, but it can change their values. The new eigenvalues will be the original eigenvalues multiplied by the corresponding diagonal elements of the diagonal matrix.

## 2. What is the significance of the diagonal elements in a diagonal matrix when it comes to eigenvalues?

The diagonal elements in a diagonal matrix determine the scaling factor for each eigenvalue when the matrix is multiplied. This means that the diagonal elements can either increase or decrease the eigenvalues of the original matrix.

## 3. Can a diagonal matrix with all zero elements have any effect on the eigenvalues of a matrix?

No, a diagonal matrix with all zero elements will have no effect on the eigenvalues of a matrix. This is because the multiplication by zero will result in all eigenvalues being equal to zero.

## 4. How does the size of a diagonal matrix affect the eigenvalues of a matrix?

The size of a diagonal matrix does not affect the eigenvalues of a matrix. As long as the diagonal matrix has the same number of rows and columns as the original matrix, the multiplication will result in the same number of eigenvalues.

## 5. Can multiplying a matrix by a diagonal matrix change the eigenvectors?

No, multiplying a matrix by a diagonal matrix will not change the eigenvectors. The eigenvectors will remain the same, but their corresponding eigenvalues may change due to the scaling effect of the diagonal elements.

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