Constructing a vector norm on $\mathbb{R}^n$ such that subordinate matrix norm equals the spectral radius

Question

Statement of problem: "Let $A$ be square diagonalizable matrix. Constructing a vector norm on $\mathbb{R}^n$ such that subordinate matrix norm, $||A||=\max|\lambda_i|$"

I know that $A$ being square diagonalizable $\implies A=PDP^{-1}$ where $P$ is invertible and $D$ is diagonal. What does this have to do with the problem? I can't see the connection

I guess my question is, what does it mean when they say to construct a vector norm?

Answer 1

When you have a norm $\|\cdot\|$ in $\mathbb{R}^n$, you can construct a matrix norm (in the space of linear operators from $\mathbb{R}^n$ to $\mathbb{R}^n$), which is given by $$ \|A\| = \sup_{x\ne 0}\frac{\|Ax\|}{\|x\|} = \sup_{\|x\|=1}\|Ax\|. $$

You are being asked to find a norm $\|\cdot\|$ such that

$$ \sup_{x\ne 0}\frac{\|Ax\|}{\|x\|} = \max|\lambda_i|. $$

note: You may want to check the Euclidean norm.