I cannot get my head around this question. I would very much appreciate any help please.
If we are given invertible matrices A, B and P so that A = PB, we can say that A is ‘left equivalent to B’. Prove that ‘left equivalence’ is an ‘equivalence relation’.
Hi,
I think you are still missing Plato's point. You need the exact definition of "left equivalence". Almost surely, the definition is:
Given invertible matrices A and B of the same size, A is left equivalent to B if and only if there is an invertible matrix P with A = PB.
So now you can prove that left equivalence is an equivalence relation on the set of all invertible $n\times n$ matrices for a given constant size $n$.
Reflexive: Let A be an invertible $n\times n$ matrix. Then A = IA (I is the identity $n\times n$ matrix). So this shows there is an invertible matrix P with A = PA, namely P = I.
You should now be able to prove the symmetric and transitive properties.
Edit: I should have added this thought. For a "meaningful" equivalence relation, let m and n be fixed positive integers and $M(n,m)$ the collection of all $n\times m$ matrices with entries from the reals (or any other field). Then for $A,\,B\in M(n,m)$, A is left equivalent to B if and only if there is an invertible $n\times n$ matrix P with A = PB.
By the way, you say "If we are given invertible matrices A, B and P so that A = PB, we can say that A is ‘left equivalent to B". That doesn't look right to me because there is no "P" in the conclusion and you appear to be saying that A and B being equivalent depends upon some given matrix "P". I suspect what you really mean is "Two matrices, A and B, are left equivalent if and only if there exist an invertible matrix, P, such that A= PB.
You need to show:
1) Reflexive. Can you find an invertible matrix, P such that A= PA?
2) Symmetric. Suppose there exist an invertible matrix, P, such that A= PB. Can you find an invertible matrix, Q, such that B= QA?
3) Transitive. Suppose there exist an invertible matrix, P, such that A= PB and an invertible matrix Q such that B= QC. Can you find an invertible matrix, R, such that A= RC?
I'll give you a big hint for proving symmetry of left-equivalent.
We will assume A is left-equivalent to B, and to show symmetry we need to prove that B is left-equivalent to A.
Since A is left-equivalent to B, we know there is an invertible matrix P with A = PB.
Since P is invertible, P^{-1} exists, so multiply both sides of the equation A = PB by it.