4
$\begingroup$

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

$\endgroup$
0

1 Answer 1

7
$\begingroup$

this is called the numerical radius of a matrix

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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