Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

OK, here is what is probably a stupid question.

Let $M$ be a non-symmetric real matrix: for example, the shear matrix $\left( \begin{array}{cc} 1 & 1 \\\ 0 & 1 \end{array} \right)$.

There are three things we might ask about it:

  1. its largest eigenvalue, which in this case is $1$.

  2. its largest singular value, or equivalently $\max_{u,v} \langle u, Mv \rangle$ where we maximize over all pairs of unit vectors $u, v$. This is also known as its operator norm; in this case it's $(\sqrt{5}+1)/2 = 1.618...$

  3. a third thing, namely $\max_v \langle v, Mv \rangle$ where we maximize over all unit vectors $v$. In other words, we restrict the previous maximization to the case $u=v$. In this case, this quantity is $3/2$.

Of course, for symmetric matrices these three things are equal. But in general they're not.

Does this third thing have a name? I'm interested in it, for instance, because if $dv/dt = M \cdot v$ I want to bound the derivative of $|v|^2$, which is $2 \langle v, Mv \rangle$.

thanks,

Cris

share|improve this question
2  
Calling Cris... are you going to accept Carlo's answer? –  Yemon Choi Jul 3 '13 at 6:46
add comment

1 Answer

this is called the numerical radius of a matrix

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.