I'm struggling to understand induced matrix p-norms. Our lecture notes have used the example:
$$A := \begin{bmatrix}\frac{3}{2} & 0 & \frac{1}{2}\\0 & 3 & 0\\ \frac{1}{2} & 0 & \frac{3}{2} \end{bmatrix}$$
In this example we're told that we have computed the induced matrix p-norms of A for $p \in \{1,2,\infty\}$ and found them to be 3.
There isn't any working out to how this was achieved, would anyone be able to go through the working.
I understand that in this case $|A|_p$ := sup $\frac{|Ay|_p}{|y|_p}$ for all $p\in [1,\infty]$ but the notes have made it very unclear as to what y is in this case, and what exactly the notation of $|Ay|_p$ means.
Best Answer
I can't tell you how you did things in class, but here is a way to compute each of those $p$-norms.
$p=1$: we want the maximum absolute column sum. That is, $$ \|A\|_1 = \max_{j} \sum_{i=1}^m |a_{ij}| $$ The first column sum is $3/2 + 1/2 = 2$. The second is $3$. The the third is $2$. So, the maximum is $3$.
$p=2$: For any symmetric matrix, $\|A\|_2$ is the maximum absolute value of the eigenvalues. In this case, the largest eigenvalue of $A$ (in absolute value) is $3$.
$p = \infty$: We want the absolute row-sum. That is, $$ \|A\|_{\infty} = \|A^T\|_1 = \max_i \sum_{j=1}^n |a_{ij}| $$ since $A$ is symmetric, we get $3$ again.