Give an example of two 2 times 2 matrices such that the oper

Give an example of two 2 times 2 matrices such that the operator norm of the product is less than the product of the operator norms.

Solution

The operator norm of the 2 by 2 matrix A = 1 2 3 4 .

Since AA> = 5 11 11 25 , the characteristic polynomial of AA> is X230X +4, whose largest eigenvalue is 15+ 221. Therefore the operator norm of A is p 15 + 221. That is, for every (x, y) R2 , ||(5x + 11y, 11x + 25y)|| q 15 + 221||(x, y)||, and p 15 + 221 is the smallest number with this property. Try computing this operator norm from the definition! To summarize, Peano’s theorem reduces the computation of an operator norm to the computation of the largest root of a (monic) polynomial whose roots are known to be nonnegative. So to program a computer to calculate an operator norm, we need an upper bound on the size of the roots of a polynomial, and such bounds exist in terms of the size of the coefficients. It is left to the reader to find such bounds in the literature or produce them anew.