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Thread: Norm of not-square matrixes

  1. #1 Norm of not-square matrixes 
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    How does definition of matrix norm (induced and not induced - Frobenius Norm, for example) change for not-square matrixes?

    In particular about:

    - submultiplicative property ||AB||≤||A||||B|| of a generic matrix norm;

    - property ||A||≥ρ(A), where ρ(A) is the greatest eigenvalue of matrix A as absolute value;


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  3. #2 Re: Norm of not-square matrixes 
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    Quote Originally Posted by doctor_cat
    How does definition of matrix norm (induced and not induced - Frobenius Norm, for example) change for not-square matrixes?

    In particular about:

    - submultiplicative property ||AB||≤||A||||B|| of a generic matrix norm;

    - property ||A||≥ρ(A), where ρ(A) is the greatest eigenvalue of matrix A as absolute value;
    The norm of an operator A between normed spaces is the supremum of the norm of
    A(x) where x ranges over elements in the domain of norm less than or equal to 1.


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  4. #3  
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    And what about the properties?
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  5. #4  
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    Quote Originally Posted by doctor_cat
    And what about the properties?
    It is a norm, with the propertiess of a norm.
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  6. #5  
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    So aren't submultiplicative property and ||A||≥ρ(A) property fundamental for a norm?

    They are valid only for square matrixes and nothing more.
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  7. #6  
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    Quote Originally Posted by doctor_cat
    So aren't submultiplicative property and ||A||≥ρ(A) property fundamental for a norm?
    your notation is not defined.

    The submultiplicative property applies to a normed algebra, not to every normed space.

    Quote Originally Posted by doctor_cat
    They are valid only for square matrixes and nothing more.
    nope

    You need square matrices for a normed algebra , but not for just a normed space.
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