Product of Hermitian matrices

  • #1
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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_k|k \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_k|k \rangle \langle k|[/tex]
[tex]B=\sum_l \lambda_l|l \rangle \langle l|[/tex]
[tex]AB=\sum_{k,l}\lambda_k\lambda_l|k \rangle \langle k|l\rangle \langle l|[/tex]
Now I am confused what to do. Definitely, ##\langle k|l \rangle \leq 1 ##. Could you help me?
 
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Answers and Replies

  • #2
StoneTemplePython
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hint: the Operator 2 Norm is submultiplicative -- find a way to use this.
 
  • #3
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Thanks. So
[tex]||AB||\leq ||A|| \cdot ||B||[/tex]
this is submultiplicativity? But which norm?
 
  • #4
StoneTemplePython
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Thanks. So
[tex]||AB||\leq ||A|| \cdot ||B||[/tex]
this is submultiplicativity? But which norm?
The Operator 2 norm-- I said this in post #2. A very quick google search should tell you that the operator 2 norm is given by ##\big \Vert C \big\Vert_2##
 
  • #5
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Sorry, but I am not sure what is Operator 2 norm.
 
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  • #6
StoneTemplePython
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  • #8
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Using Frobenius norm I know that
[tex]||A||=\sqrt{(A,A)}=\sqrt{Tr(A^*A))}[/tex]
and
[tex]||AB||\leq ||A|| ||B|| [/tex]
[tex]|\lambda_A|\leq ||A||[/tex]
[tex]|\lambda_B|\leq ||B||[/tex]
[tex]|\lambda|\leq ||AB||[/tex]
but I am still not sure how to prove that ##|\lambda|>1##.
 
  • #9
StoneTemplePython
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no. you need to use the operator 2 norm. The Frobenius norm isn't the right tool for this job. Do you know what a singular value is? If not, this problem may be out of reach.
 
  • #10
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I checked it on wikipedia
Something like
##AX_k=\sigma_kY_k##
##A^*Y_k=\sigma_kX_k##
but it is hard to me to understand how to find ##\sigma_k## on the concrete example.
 

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