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Homework Statement:

For two Hermitian matrices ##A## and ##B## with eigenvalues larger then ##1##, show that
##AB## has eigenvalues ##\lambda>1##.
Relevant Equations:

Any hermitian matrix ##A## could be written as
[tex]A=\sum_k \lambda_kk \rangle \langle k[/tex]
where ##k\rangle \langle k## is orthogonal projector ##P_k##.
Product of two Hermitian matrix ##A## and ##B## is Hermitian matrix only if matrices commute ##[A,B]=0##. If that is not a case matrix ##C=AB## could have complex eigenvalues. If
[tex]A=\sum_k \lambda_kk \rangle \langle k[/tex]
[tex]B=\sum_l \lambda_ll \rangle \langle l[/tex]
[tex]AB=\sum_{k,l}\lambda_k\lambda_lk \rangle \langle kl\rangle \langle l[/tex]
Now I am confused what to do. Definitely, ##\langle kl \rangle \leq 1 ##. Could you help me?
[tex]A=\sum_k \lambda_kk \rangle \langle k[/tex]
[tex]B=\sum_l \lambda_ll \rangle \langle l[/tex]
[tex]AB=\sum_{k,l}\lambda_k\lambda_lk \rangle \langle kl\rangle \langle l[/tex]
Now I am confused what to do. Definitely, ##\langle kl \rangle \leq 1 ##. Could you help me?