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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 $2 \langle v, Mv \rangle$.



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Calling Cris... are you going to accept Carlo's answer? – Yemon Choi Jul 3 '13 at 6:46

this is called the numerical radius of a matrix

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