Suppose that $T$ is a normal operator on $V$. Show that $\|T(v+w)\|=10$.

Question

Suppose $T$ is a normal operator on $V$. Suppose also that $v, w \in V$ satisfy the equations $$ \| v \|= \| w \| =2, Tv = 3v, Tw = 4w.$$ Show that $\| T(w+v) \| = 10.$

I thought this problem would be easy, but I am stuck. I began by writing \begin{align*} \|T(v+w)\|&=\sqrt{\langle T(v+w),T(v+w)\rangle}\\ &=\sqrt{\langle Tv,Tv\rangle+\langle Tv,Tw\rangle+\langle Tw,Tv\rangle+\langle Tw,Tw\rangle} \end{align*} At this point I figure I should use the fact that $T$ is normal (i.e., $TT^*=T^*T$, where $T^*$ is the adjoint of $T$), but I just can't figure out how! Hints would be appreciated.

Answer 1

First, recall or prove that for a normal operator eigenvectors of two distinct eigenvalues are orthogonal.

Second, do as you did, then replace $Tv$ and $Tw$ by $3v$ and $4w$, respectively, pull out the constants, and simplify.