A question on Matrix Norms

Question

Suppose $\mathbf{A}\in\mathbb{R}^{n\times n}$, and $\mathbf{B}\in\mathbb{R}^{n\times n}$ be a matrix with strictly positive entries. Define ${\Vert\mathbf{A}\Vert}_{\mathbf{B}}=\max\limits_{1\leq i,j \leq n} b_{ij}|a_{ij}|$. Then, is ${\Vert\mathbf{A}\Vert}_{\mathbf{B}}$ a matrix norm? Other than the sub-multiplicative property, I have been able to verify all the other requirements of a matrix norm.

Answer 1

Not for arbitrary $B$. For example, for $B=\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, your definition of $\| A \|_B $ equals the maximum entry of $A$. For $C=\begin{pmatrix} 10 & 1 \\ 0 & 1 \end{pmatrix}$, $D=\begin{pmatrix} 1/10 & 1 \\ 0 & 1 \end{pmatrix}$, you now have $\| CD \|_B = 11$, while $\| C \|_B = 10$ and $\| D \|_B = 1$.