# Equivalence Matrices

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• Dec 4th 2009, 06:36 PM
kjchauhan
Equivalence 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..
• Dec 4th 2009, 06:51 PM
Bruno J.
What do you want exactly? All matrices equivalent to $A$?
• Dec 4th 2009, 07:09 PM
kjchauhan
Any one..
suppose the matrix $A=\left[\begin{matrix} 1 & 2 & 3\\ 1 & 0 & 2\\ 3 & -1 & 0 \\\end{matrix}\right]$.
Then how we can find an equivalent matrix of A?
• Dec 4th 2009, 07:11 PM
lvleph
Well, the easiest and dumbest answer is to let $P=Q=I \text{ and } B = A$. One way to find a similar matrix is by making the matrix B be a matrix with the eigenvalues of $A$ as the diagonal of $B$. Then the matrix $P$ would have the corresponding eigenvectors as columns and $Q = P^{-1}$. That is, if $(\lambda_i, v_i)$ is an eigenpair then
$B = \begin{pmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \cdots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & \cdots & 0 & \lambda_n\end{pmatrix}$
and
$P = \begin{pmatrix} v_1 \cdots v_n\end{pmatrix}$.
Somehow I doubt this is what you are looking for.
• Dec 4th 2009, 07:17 PM
kjchauhan
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. $A=P^{-1}BP$ and for equivalent matrices we have two nonsingular matrices s.t. $A=PBQ$..
• Dec 4th 2009, 07:26 PM
lvleph
Well, similar matrices are equivalent by definition. Since $P,P^{-1}$ are non-singular and taking $P=P,Q=P^{-1}$ we fulfill the definition.
• Dec 5th 2009, 05:51 AM
HallsofIvy
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 $PAP^{-1}$.

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 $D= P^{-1}AP$ is a diagonal matrix equivalent to A.