When does exist matrices T and H such that HCE=TE? (all matrices are rectangular)

Question

could you please help me with this question; I want to find out the conditions (necessary and /or sufficient) for the existence of two matrices namely H and T such that the equality HCE=TE holds for given matrices C and E. All matrices are rectangular. What about a simpler case would be when E is a column vector? thanks a lot.

Answer 1

For the case where $E$ is a vector.

Well if $E$ is a vector then $CE$ is another vector. So your question is when there exists two matrices $T,H$ which sends one vector to another. The solution is always, unless one of the vectors is $0$.

If $E$ is the zero vector then any matrices $T,H$ would satisfy the equation.

If $E$ is non-zero but $CE$ is zero then we can take $T$ to be the zero matrix.

If non of them is zero then we can choose any matrix $T$, find a basis which consists $CE$ and then simply find a matrix which sends $CE$ to $TE$ and the rest of the vectors to zero.

Conclusion: if $E$ is a column vector the existence of $T,H$ is guaranteed with no additional conditions.

Some notes on the general case: If $E$ is a matrix, then denote by $E_1,E_2,...,E_n$ it's colum vectors. Then the column of $TE$ are $TE_1,TE_2,...,TE_n$. So the question above is basically whether we can find $T,H$ for multiple vectors $E_1,...,E_n$ simultaneously and so I believe it also works in the general case (i.e you can always find such $T,H$).