# Similarity Matrices

• Dec 4th 2010, 02:30 AM
ercan
Similarity Matrices
I need help about similarity transformation in matrices.
Is there anyone who knows how can I decide whether "the two matrices having the same eigenvalues" are similar or not without using eigenvectors.

For example, following two matrices have the same characteristic polynomial. But they are not similar. Because matrix A has multiple jordan blocks for the double eigenvalue 1 while B doesn't have multiple jordan block.

A=[3 -3 -1 2;
4 -2 -3 6;
4 -3 -2 6;
3 0 -3 7]

B=[-4.6 -7 -3 -0.6;
3.4 6 2 0.4;
0.4 0 1 0.4;
-2.4 5 0 3.6]

I wonder how can I show that A and B are not similar without using their eigenvectors or without directly looking to their diagonal forms.
• Dec 4th 2010, 03:30 AM
tonio
Quote:

Originally Posted by ercan
I need help about similarity transformation in matrices.
Is there anyone who knows how can I decide whether "the two matrices having the same eigenvalues" are similar or not without using eigenvectors.

For example, following two matrices have the same characteristic polynomial. But they are not similar. Because matrix A has multiple jordan blocks for the double eigenvalue 1 while B doesn't have multiple jordan block.

A=[3 -3 -1 2;
4 -2 -3 6;
4 -3 -2 6;
3 0 -3 7]

B=[-4.6 -7 -3 -0.6;
3.4 6 2 0.4;
0.4 0 1 0.4;
-2.4 5 0 3.6]

I wonder how can I show that A and B are not similar without using their eigenvectors or without directly looking to their diagonal forms.

As far as I can see right now that's impossible: you need either the eigenvectors and their eigenspaces' dimensions, or what ammounts

to way more work and you ALSO need eigenvectors and etc. , their Jordan Canonical Forms.

I can't see any short cut for this in general. Now, for matrices of size up to 3x3, it's enough to know their characteristic and minimal

polynomials, but for that...

Tonio