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Information about the maximum

  1. Sep 8, 2009 #1
    Does anybody have a good book/website where I can find good information on how to use the maximum on matrices. I have to prove an expression involving the maximum and eigenvalue of matrices. But I don't know how to link those to together. I think I can figure this out, if only I had some good information source :)

    Regards, Azizz
     
  2. jcsd
  3. Sep 8, 2009 #2

    morphism

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    What do you mean by "the maximum"?
     
  4. Sep 8, 2009 #3
    Sorry I went to fast here. With the maximum I meant the largest (or maximal) eigenvalue, for example

    [tex] \lambda_{\max}(A) = \max_{\| x \| =1} x^* A x [/tex]

    Then my question is: what do I know of this operator? Is it, eg, linear?
     
  5. Sep 8, 2009 #4

    morphism

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    You mean is the function [itex]\lambda_\max[/itex] linear on the space of matrices? Certainly not.
     
  6. Sep 8, 2009 #5
    Ok, but I think this holds true:

    Suppose A-B is hermitian and positive definite, then

    [tex] \max_{\|x\|=1} x^*(A-B)x \geq x^*(A-B) x = x^*Ax - x^*Bx \leq \lambda_{\max}(A) - \lambda_{\max}(B) [/tex]
     
  7. Sep 9, 2009 #6
    Found partly what I needed:

    [tex] \lambda_{\max}(A)I \geq A \geq \lambda_{\min}(A)I [/tex]

    [tex] \beta I > A \iff \beta > \lambda_{\max}(A) [/tex]

    Now all I have to know is what is known for the eigenvalue of two matrices? That is:

    [tex] \lambda_{\max}(A+B) = ... [/tex]

    Is there any expression I can use for such an equality (or perhaps inequility)?
     
    Last edited: Sep 9, 2009
  8. Sep 11, 2009 #7

    morphism

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    I don't think you can say anything intelligent for arbitrary matrices A and B. (But I could be wrong!)
     
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