Norms of vectors and matrices
Norms
Consider a linear system \(Ax = b\) where \(A,b\) are input.
To measure the conditioning we measure the norm of output
Claim \(\|x\|_\infty \leq \|x\|_2\leq \sqrt n \|x\|_\infty\)
proof.
Claim reverse triangular inequality \(|\|x\| - \|y\|| \leq \|x-y\|\)
Norm for matrices
One common way to get a matrix norm is from a vector norm \(\|A\| = \max_{x\neq 0} \frac{\|A\vec x\|}{\|\vec x\|}\), i.e. how much \(A\) can expand \(\vec x\).
Theorem \(\|A\| = \max_{x\neq 0} \frac{\|Ax\|}{\|x\|} = \max_{\|x\|=1} \|Ax\|\)
proof.
Matrix norm is induced by / subordinate to the vector norm,
Examples Given \(A_{n\times n}\)
Note that F-norm cannot be induced by vector norm. For example $|I|_F = \sqrt n $
If a norm is subordinated from a vector norm, then - it follows triangular inequality
- \(\|I\| = \max \frac{\|Ix\|}{\|x\|} = 1\)
Norm Equivalence
For any vector and matrix norms \(\|\cdot\|_a, \|\cdot\|_b\),