Similar Diagonal Matrices

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In summary, the conversation discusses a question regarding the classification of intertwining operators of two group representations. The question asks for the specific matrices that satisfy the equality AXA^{-1} = X, with X being an n \times n diagonal matrix with n distinct non-zero eigenvalues. It is determined that the answer to this question is the set of operators with the same invariant subspaces, where the eigenvalues do not necessarily have to be the same, but must be simultaneously diagonalizable.
  • #1
jgens
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As part of a larger problem involving classifying intertwining operators of two group representations, I came across the following question: If [itex]X[/itex] is an [itex]n \times n[/itex] diagonal matrix with [itex]n[/itex] distinct non-zero eigenvalues, then exactly which [itex]n \times n[/itex] matrices [itex]A[/itex] satisfy the following equality [itex]AXA^{-1} = X[/itex]? Does anyone know the answer to this question?

Edit: Nevermind. I found a better way of doing the problem that avoids this sort of argument.
 
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  • #2
Those whose eigenvalues are the numbers on the diagonal of the original matrix.
 
  • #3
Is that true? I believe it is the set of operators with the same invariant subspaces. The eigenvalues don't have to be the same, they just have to be simultaneously diagonalizable.
 

1. What is a similar diagonal matrix?

A similar diagonal matrix is a type of square matrix that has the same diagonal elements as another matrix, but with different values for the off-diagonal elements. This means that the two matrices have the same eigenvalues, but different eigenvectors.

2. How do you determine if two matrices are similar diagonal matrices?

To determine if two matrices are similar diagonal matrices, you can check if they have the same diagonal elements and if their eigenvectors are different. If both conditions are met, then the matrices are similar diagonal matrices.

3. What are the properties of similar diagonal matrices?

Similar diagonal matrices have the same determinant, trace, and rank. They also have the same eigenvalues, but different eigenvectors. Additionally, they have the same diagonalization and diagonal form.

4. How are similar diagonal matrices useful in linear algebra?

Similar diagonal matrices are useful in linear algebra because they represent a change of basis. This can be helpful in solving systems of linear equations and understanding transformations in vector spaces.

5. Can a matrix be similar to itself?

Yes, a matrix can be similar to itself. This is known as a diagonalizable matrix, where the matrix is similar to a diagonal matrix with its own eigenvalues on the diagonal. It is also known as a scalar matrix, where the off-diagonal elements are all zero.

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