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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!
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.
