Effect on eigenvalues of multiplying by a diagonal matrix

Click For Summary
SUMMARY

The discussion focuses on the effect of multiplying a matrix by a diagonal matrix on its eigenvalues, specifically in the context of a MIMO linear precoder optimization problem. The matrix A is known, and the diagonal matrix D contains the optimization variables. The transformation of the eigenvalues is explored through the diagonalization of the matrix B, defined as A^HA. The participants suggest that diagonalizing B using a transformation matrix S allows for a clearer understanding of how the eigenvalues are affected by the multiplication with D.

PREREQUISITES
  • Understanding of eigenvalues and eigenvectors
  • Familiarity with MIMO (Multiple Input Multiple Output) systems
  • Knowledge of matrix diagonalization techniques
  • Experience with optimization problems in linear algebra
NEXT STEPS
  • Study the properties of eigenvalues in the context of diagonal matrices
  • Learn about MIMO linear precoding techniques and their optimization
  • Explore matrix diagonalization methods and their applications
  • Research the implications of transforming matrices on eigenvalue distributions
USEFUL FOR

Researchers and engineers working in communications, particularly those involved in MIMO system design and optimization, as well as students studying linear algebra and matrix theory.

Raito
Messages
2
Reaction score
0
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.
 
Physics news on Phys.org
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
 

Similar threads

Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
3K
Undergrad Spectral theorem for Hermitian matrices-- special cases
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Eigenstates of "summed" matrix
  • · Replies 1 ·
Replies
1
Views
2K
Diagonalizing a matrix given the eigenvalues
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Is a symmetric matrix with positive eigenvalues always real?
  • · Replies 8 ·
Replies
8
Views
3K
Graduate Solutions of Matrix Equation A X B^T = C
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Understanding Diagonalization and Eigenvalues in Matrix Transformations
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Parameter optimization for the eignevalues of a matrix
  • · Replies 0 ·
Replies
0
Views
2K
Undergrad General form of symmetric 3x3 matrix with only 2 eigenvalues
  • · Replies 4 ·
Replies
4
Views
6K
Graduate Exponent of matrix/Diagonalization of matrix with repeated eigenvalue
  • · Replies 1 ·
Replies
1
Views
3K