Source for if $L_A:(\mathbb{R}^n, \left\Vert \cdot \right\Vert_p) \to (\mathbb{R}^n, \left\Vert \cdot \right\Vert_q)$ is an isometry, then $p = q$.

Question

Let and $n\geq 3$ be a natural number and $p,q\in [1,\infty]$.

For any $n\times n$ square matrix $A$, let $L_A:(\mathbb{R}^n, \left\Vert \cdot \right\Vert_p) \to (\mathbb{R}^n, \left\Vert \cdot \right\Vert_q)$ be defined by $L_A(z) = Az$, that is, $L_A$ is the pre-matrix multiplication operator.

I think the following fact is well-known.

If $L_A$ is an isometry, then $p = q$.

I am looking for a research paper or book that cites the result above. However, I am not able to get any.