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Definition of a complex matrix

  1. Sep 11, 2008 #1
    MathWorld states that a complex matrix is "a matrix whose elements may contain complex numbers". My question is what the "may" means. Could a matrix be complex even if its elements does not contain any complex numbers?
     
  2. jcsd
  3. Sep 11, 2008 #2

    Dick

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    Real matrices are contained in the set of complex matrices. In that sense they are also complex. That's the only sense in which they are complex. A real matrix is, uh, real.
     
  4. Sep 12, 2008 #3
    Thanks. I have a follow-up question if you don't mind: does this mean that an element in the [tex]\mathbb{R}[/tex]-vector space [tex]Herm_n(\mathbb{C})[/tex] (the set of all hermitian n x n-matrices) is also an element in [tex]\mathbb{C}[/tex]^n?
     
    Last edited: Sep 12, 2008
  5. Sep 12, 2008 #4

    Dick

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    If you mean is the set of all hermitian matrices a subset of the set of all complex matrices, of course it is. Why do you need to ask?
     
  6. Sep 12, 2008 #5
    Because I'm trying to solve a problem in which there is an expression [tex]AB[/tex], where [tex]A[/tex] is a [tex]\mathbb{C}[/tex]-linear map [tex]A:\mathbb{C}^n\rightarrow\mathbb{C}^n[/tex] and [tex]B \in \msbox{Herm_n}(\mathbb{C})[/tex]. If [tex]B[/tex] doesn't also lie in [tex]\mathbb{C}^n[/tex], I can't make sense of the expression.
     
    Last edited: Sep 12, 2008
  7. Sep 12, 2008 #6
    Adding to my confusion was this statement on Wikipedia:
     
  8. Sep 12, 2008 #7

    Hurkyl

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    What's wrong with reading it as simply being the product of linear operators?
     
  9. Sep 12, 2008 #8

    Dick

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    We have agreed Herm_n(C) does lie in the set nxn complex matrices, right? It's not a vector space over C because if you multiply a hermitian matrix by i it's no longer hermitian. But I don't see why that needs to concern you.
     
  10. Sep 14, 2008 #9
    Yes.

    Very helpful! Thanks.

    I've actually solved the problem, and you're right, it was unnecessary for me to be concerned by it.

    Thanks again for your help.
     
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