The size of two similar matrices

Question

Show that if B is similar to A, then they are both square matrices of the same size.

If someone could get me started in the right direction, that would be helpful.

Answer 1

I'm used to the definition of similar matrices explicitly requiring the matrices to be square. But, I suppose it's not strictly necessary. I think the essence of saying that matrices $A_1$ and $A_2$, of respective sizes $m_1 \times n_1$ and $m_2 \times n_2$, is that there exists an invertible $P$ such that, $$A_1 = P^{-1} A_2 P$$ Let $m_3 \times n_3$ be the size of $P$. The first thing to realise is that $P$ must be square in order to be invertible (for our purposes, we say $Q$ is an inverse of $P$ if $PQ$ and $QP$ are identity matrices of possibly different sizes). If we think about this in terms of linear maps, this means $P$ represents an invertible linear map from $\mathbb{R}^{m_3}$ to $\mathbb{R}^{n_3}$, which is famously only possible if $m_3 = n_3$. Thus, $P$ must be square, as claimed.

For $P^{-1}A_2$ to be well-defined, we must have $n_3 = m_2$. For $A_2 P$ to be well-defined, we must have $n_2 = m_3$. Putting altogether, $$n_2 = m_3 = n_3 = m_2,$$ so $A_2$ and $P$ are all square, and all the same sizes. Thus, so is $A_1$.