# 2-norm and infinty norm of a system in controls

How to compute 2-norm or infinity norm of following system?

i am confused whether to calculate using simple matrix theory "where it don't regard for s domain" or H2 and H-infinty norm. Thanks in advance. The problem is 4.4 from book Essentials of Robust Control

You probably know that matrix norms can be defined by the vector norms in the following way: $$$$||A||:= \max_{x\neq 0} \frac{||Ax||}{||x||}$$$$ for a matrix $$A$$. So you just look up the definition for the infinity or 2-norm and plug it into the expression above. When doing this you will further realize that the following equalities hold: $$$$||A||_\infty = \max\limits_i \sum\limits_{j}|a_{ij}|\\ ||A||_2= \sqrt{\lambda_{max}(A^HA)}$$$$ So for the infinity norm you sum up the absolute values in each row and take the the row with the largest sum. For the 2-norm you need to find the eigenvalue with the largest absolute value for the matrix $$A^HA$$. Note when you have the variable $$s$$ in your matrix the norms might also be functions of $$s$$.