Q.
On the set M of all $m * n$ real matrices, introduce the following equivalence relation: two such
matrices A and B are equivalent if there exist a real invertible $m * m$ matrix P and a real invertible
$n * n$ matrix Q such that A = PBQ. Into how many disjoint equivalence classes is M partitioned
by this equivalence relation?
I think
To determine the number of disjoint equivalence classes, we need to find how many different equivalence classes exist in M.
Consider the matrix A .The matrix A can be decomposed as $A = P_1DQ_1$, where $P_1$ is an invertible matrix, D is a diagonal matrix with diagonal elements λ1, λ2, …, λr, where r is the rank of A, and $Q_1$ is an invertible matrix.
Similarly, any matrix B in M can be decomposed as $B = P_2DQ_2$, where $P_2$ is an invertible matrix, D is a diagonal matrix with diagonal elements μ1, μ2, …, μs, where s is the rank of B, and $Q_2$ is an invertible matrix.
If A and B are equivalent, then there exist invertible matrices P and Q such that A = PBQ. Substituting the expressions for A and B, we get $P_1DQ_1$ = $PBPQ_2$. Since $P_1$ and $P_2$ are invertible, we can left-multiply both sides by $P_2^-1$ and right-multiply by $Q_1^-1$ to get $D = P_2^-1BPQ_1^-1.$
Thus, two matrices A and B are equivalent if and only if they have the same rank and the diagonal entries of the diagonal matrices D in their respective decompositions are equal up to permutation.
Therefore, the number of disjoint equivalence classes in M is equal to the number of different sets of diagonal entries of rank r matrices, where r ranges from 0 to min(m, n).
Am i in right way or anything mistake.. please help me.
Best Answer
This equivalence relation is actually called "matrix equivalence." Two matrices of the same dimensions are equivalent if and only if they have the same rank (no other condition is required), so for the set of $\ m\times n\ $ real matrices, there are $\ \min(m,n)+1\ $ equivalence classes, corresponding to each of the possible ranks $\ 0,1,2,\dots,\min(m,n)\ $.