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Thread: Matrix Eigenvalues

  1. #1 Matrix Eigenvalues 
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    Can anybody please tell me what is the real physical meaning of an eigenvalue of any square matrix?


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    Pritish
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  3. #2  
    Forum Radioactive Isotope MagiMaster's Avatar
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    That depends heavily on the real physical meaning of the matrix.


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  4. #3  
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    give me an example
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    Pritish
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  5. #4  
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    well, i want to know what was he motivation behind the "invention" of eigenvalue concept
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    Pritish
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  6. #5  
    Forum Radioactive Isotope MagiMaster's Avatar
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    I can't really give any concrete examples, but I can at least point to this.
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  7. #6  
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    I can, I think, though I am way off my patch here.

    It seems to be an axiom of quantum physics that any observable event in nature can be represented by a linear operator on some vector space, and that the result of any physical measurement of the action of on this space, specifically an Hilbert space , will by an eigenvalue for the operator , Or rather, since we are the quantum world, the spectrum (not necessarily non-degenerate) of eigenvalues for give the range of discrete values for these measurements,

    Back on my own patch: it is a nice (and not difficult) proof that the eigenvalues of an Hermitian operator (the physicists choice of operator) are always real (try it!), which makes complete sense with the above: what is meant by an imaginary measurement???
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  8. #7  
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    Thanks guys. I didn't quite understand the quantum mech. definition, but the wikipedia link helped.
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    Pritish
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  9. #8  
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    Quote Originally Posted by PritishKamat
    Thanks guys. I didn't quite understand the quantum mech. definition, but the wikipedia link helped.
    Be careful here. In quantum mechanics you are not working with a matrix but with a linear operator, often unbounded. The notion of a spectrum in that case becomes quite a bit more complicated.

    In finite dimensions all elements of the spectrum are eigenvalues. And all matrices are representations of bounded (continuous) linear operators, once a basis has been fixed. In that case an eigenvector is one on which the linear transformation works as a simple stretching, contraction or reversal and the eigenvalue is the scale factor.
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