Please illustrate and show me that how to find equivalence matrix of a given matrix A.

Two Matrices A and B are called equivalence if for any two non-singular matrices P and Q such that, A=PBQ.

Pl help..

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- December 4th 2009, 05:36 PMkjchauhanEquivalence Matrices
Please illustrate and show me that how to find equivalence matrix of a given matrix A.

Two Matrices A and B are called equivalence if for any two non-singular matrices P and Q such that, A=PBQ.

Pl help.. - December 4th 2009, 05:51 PMBruno J.
What do you want exactly? All matrices equivalent to ?

- December 4th 2009, 06:09 PMkjchauhan
Any one..

suppose the matrix .

Then how we can find an equivalent matrix of A? - December 4th 2009, 06:11 PMlvleph
Well, the easiest and dumbest answer is to let . One way to find a similar matrix is by making the matrix B be a matrix with the eigenvalues of as the diagonal of . Then the matrix would have the corresponding eigenvectors as columns and . That is, if is an eigenpair then

and

.

Somehow I doubt this is what you are looking for. - December 4th 2009, 06:17 PMkjchauhan
But i think it is dioganalised matrix..

Can we take similar marices as equivalent matrices?

As I know for similar matrices we have a nonsingular matrix P s.t. and for equivalent matrices we have two nonsingular matrices s.t. .. - December 4th 2009, 06:26 PMlvleph
Well, similar matrices are equivalent by definition. Since are non-singular and taking we fulfill the definition.

- December 5th 2009, 04:51 AMHallsofIvy
You said before that you wanted "any" matrix equivalent to A. As Ivleph said, take any (invertible) 3 by 3 matrix, P, find [tex]P^{-1}[/math ]and calculate .

If you specifically want a**diagonal**matrix equivalent to A, as you seem to be saying now, you must first prove that such at thing exists- that A is "diagonalizable" which is true if and only if A has 3 independent eigenvectors. Once you have found those eigenvectors, form the matrix P having the eigenvectors as columns. The is a diagonal matrix equivalent to A.