Operator and matrix norms

Measuring the size of a matrix

This is the main norm we will use. It can be derived from the vector norms and extended to matrices

$$\Vert A{\Vert }_p=\underset{x\ne 0}{\sup }\frac{\Vert Ax{\Vert }_p}{\Vert x{\Vert }_p}=\underset{\Vert x{\Vert }_p=1}{\max }\Vert Ax{\Vert }_p$$

Example: $A=\left(\begin{array}{cc}2& 1\\ 0& 1\end{array}\right)$

2-norm / 1-norm / $\infty$-norm:

Other norms that are useful.

Frobenius norm: it is very convenient to analyze matrices that can be decomposed into blocks

$$\Vert A{\Vert }_F=(\sum _{ij}{a}_{ij}^2{)}^{1/2}$$

Useful inequalities for proofs

$$\Vert Ax{\Vert }_p\le \Vert A{\Vert }_p\Vert x{\Vert }_p$$

This is a consequence of the definition of the norm. Exercise: prove this result using the definition.

For Frobenius and $p$-norms:

$$\Vert AB\Vert \le \Vert A\Vert \Vert B\Vert$$

This is called a sub-multiplicative norm.

Vectors and matrices, Vector norms, Matrix-vector and matrix-matrix product