3
$\begingroup$

Let $X ∈ M(n, \Bbb C)$ and $\{u_1, u_2, . . . , u_n\}$ be an orthonormal basis of $\Bbb C^n$. Prove that $\lVert X\rVert^2 =\sum_{i,j=1}^\infty \lvert\langle u_i,Xu_j\rangle\rvert^2$. Here $\lVert X\rVert^2=\sum_{i,j=1}^\infty X_{ij}^2$.

I have tried like this: $\exists P$ orthogonal (as $\{u_1, u_2, . . . , u_n\}$ and $\{e_1, e_2, . . . , e_n\}$ both are orthonormal) such that $P(e_i)=u_i$ where $e_i$ is our standard basis element in $\Bbb C^n$.

Now $\sum_{i,j=1}^\infty\lvert\langle u_i,Xu_j\rangle\rvert^2=\sum_{i,j=1}^\infty\lvert\langle Pe_i,XPe_j\rangle\rvert^2=^?\sum_{i,j=1}^\infty\lvert\langle e_i,Xe_j\rangle\rvert^2$. $\ldots (1)$

Because I know that $\lvert\langle Pe_i,Pe_j\rangle\rvert=\lvert\langle e_i,e_j\rangle\rvert$ for othogonal matrix $P$ but here we have $\lvert\langle Pe_i,XPe_j\rangle\rvert$ in equation $(1)$.

I need justification.

$\endgroup$
  • 1
    $\begingroup$ I'm the \langle \rangle fairy, here to let you know that $\langle, \rangle$ play nicer with TeX than <, > do :) $\endgroup$ – Patrick Stevens Feb 2 '18 at 8:35
  • $\begingroup$ You have n basis vectors and your sum requires an infinite amount of vectors? $\endgroup$ – Raito Feb 2 '18 at 8:56
1
$\begingroup$

First use the fact that for any vector $v$, $||v||^{2}= \sum _{i=1} ^{\infty} |\langle u_i ,v \rangle|^{2}$. We now have to show that $||X||^{2}= \sum_j ||X u_j||^{2}$. Now $\sum_j ||X u_j||^{2}=\sum _{j,k} |\langle X u_j , e_k \rangle|^{2} =\sum _{j,k} |\langle u_j , Y e_k \rangle|^{2}$ where Y is the adjoint of X. Hence $\sum_j ||X u_j||^{2}=\sum ||Ye_k||^{2}$. In order to replace Y by X use the following idea: in above identities we only used the facts that $\{u_j\}$ and $\{e_j\}$ were orthonormal bases. We can interchange teh roles of these and also replace X by Y. If you do this and note that the adjoint of Y is X you will see that $\sum ||Ye_k||^{2}=\sum ||X e_k||^{2}=||X||^{2}$. I have used operator theoretic notations and I hope that is OK with you.

$\endgroup$
  • $\begingroup$ Why $||v||^2=\sum_{i=1}^\infty |<u_i,v>|^2$? and why $\sum _{j,k} |\langle u_j , Y e_k \rangle|^{2}=\sum ||Ye_k||^{2}$ $\endgroup$ – user152715 Feb 2 '18 at 9:58
  • $\begingroup$ I think if you explain first question of mine then second comes from that. $\endgroup$ – user152715 Feb 2 '18 at 10:42
  • $\begingroup$ @user152715 the first equation is a basic property of orthonormal bases and the second is a special case. I do not know how familiar you are with linear operators and orthonormal bases but I will try to clarify your doubts as far as possible $\endgroup$ – Kavi Rama Murthy Feb 2 '18 at 12:24

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.