6
$\begingroup$

Suppose $A$ and $B$ are two positive definite matrices such that $\|A+B\| = \|A\|+\|B\|$. Show that $A$ and $B$ have a common eigenvector, where $\|A\|$ is the operator norm of $A$.

I'm wondering if there is a general strategy to solving problems of this form, I've been struggling with the problems for a while and can't get it. Any help is appreciated, Thank you!

$\endgroup$
6
$\begingroup$

Suppose $A$ and $B$ are positive operators on a finite-dimensional inner product space $V$ and $\|A + B\| = \|A\| + \|B\|$.

Because $A$ and $B$ are positive, $A+B$ is also a positive operator. Thus there exists $x \in V$ such that$$\|x\| = 1 \quad \text{and} \quad\|A+B\| = \langle (A+B)x, x \rangle. $$ Now \begin{align*} \|A+B\| &= \langle (A+B)x, x \rangle\\[3pt] &= \langle Ax, x \rangle + \langle Bx, x \rangle\\[3pt] &\le \|A\| + \|B\|\\[3pt] & = \|A+B\|. \end{align*} Because the first term above equals the last term above, the inequality must be an equality. In other words, $$ \langle Ax, x \rangle = \|A\| \quad \text{and} \quad \langle Bx, x \rangle = \|B\|. $$ The equations above imply that $x$ in an eigenvector of $A$ and $x$ is an eigenvector of $B$. Hence we have shown that $A$ and $B$ have a common eigenvector, as desired.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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