Similarity Transformation of Matrices: Decide w/o Eigenvectors

[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.

 

[micromass]

The minimal polynomial could be what you're looking for...

Basically, given a matrix A, then the characteristic polynomial p(x) satisfies p(A)=0. (this is the Cayley-Hamilton theorem). The smallest polynomial which still has this property is called the minimal polynomial. The minimal polynomial will always divide the characteristic polynomial.

The minimal polynomial is mostly being used to give information about the Jordan normal form and it has the advantage that you don't need to know about the (generalized) eigenvectors...

 

[brydustin]

Go check the book: Matrix Analysis by Horn and Johnson, it answers this question exactly, or will provide sufficient information.