Suppose $A\in \mathbb{C}^{n\times m}$ and let the matrix $B$ be any submatrix of $A$. Show $\|B\|_p \leq \|A\|_p$.
We have that $$\|A\|_p = \max_{\|x\|_p = 1}\|Ax\|_p \ \ \ \text{and} \ \ \ \|B\|_p = \max_{\|x\|_p = 1}\|Bx\|_p$$ Since $B$ is a submatrix of $A$ it seems obvious that $\|B\|_p \leq \|A\|_p$ but I am not sure how to show this is true rigorously. Any suggestions is greatly appreciated.
Hint: We can find matrices $P$ and $Q$ for which $B = PAQ$ for which $\|P\|$ and $\|Q\|$ are $1$. It follows that $$ \|B\| = \|PAQ\| \leq \|P\|\,\|A\|\,\|Q\| = \|A\| $$
For example, for a $3 \times 3$ matrix $A$, we find that $$ \pmatrix{1&0&0\\0&0&1} A \pmatrix{1&0\\0&1\\0&0} = \pmatrix{a_{11}&a_{12}\\a_{31}&a_{32}} $$ In general: if $A$ is $m \times n$ the rows of $P$ are rows of the size-$m$ identity matrix, and the columns of $Q$ are columns of the size-$n$ identity matrix.