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Math Help - How to calculate the inverse of an infinite matrix?

  1. #1
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    How to calculate the inverse of an infinite matrix?

    Firstly, do we have a definition for the inverse of an infinite matrix? If we have, secondly, how to calculate it? I guess we may approximate it by inverses of finite matrices whose size is increasing larger and larger. But it's just my guess, I can't give a thorough analysis on this approximation, and I think there may be methods that can get the inverse directly, just like finite matrices. I have no idea on this problem, could you please give me a detailed exposition and analysis on this problem? Any suggestion or books on this subject is welcomed. Thanks!
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  2. #2
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    I don't know what you mean by infinite matrix. Do you just mean an nxn matrix, where n can be any number? That would just mean you want to generalize the method for calculating an inverse.
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  3. #3
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    Quote Originally Posted by zzzhhh View Post
    Firstly, do we have a definition for the inverse of an infinite matrix? If we have, secondly, how to calculate it? I guess we may approximate it by inverses of finite matrices whose size is increasing larger and larger. But it's just my guess, I can't give a thorough analysis on this approximation, and I think there may be methods that can get the inverse directly, just like finite matrices. I have no idea on this problem, could you please give me a detailed exposition and analysis on this problem? Any suggestion or books on this subject is welcomed. Thanks!
    i think by "infinite matrices" you mean "discrete infinite matrices", i.e. \aleph_0 \times \aleph_0 matrices with entries from a \mathbb{C}. of course we can define the inverse: A is called invertible if there exists

    a "unique" matrix B such that AB=BA=I. you need to be very careful when dealing with infinite matrices because:

    1) obviously the product of two infinite matrices is not necessarily defined. thus we even don't have "associativity".

    2) an infinite matrix might have a left (right) inverse but not a right (left) inverse. even if it has both, they might not be equal or even unique.

    3) the concept of "determinant" is not defined for an infinite matrix. so there is no criteria for an infinite matrix to be "invertible" except the definition.
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    Thank you for the detailed reply. Then for any given infinite matrix, we do not konw if it is invertible, not to mention how to calculate it, right? Could you tell me which branch of math this problem belongs to?
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  5. #5
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    Quote Originally Posted by zzzhhh View Post
    Could you tell me which branch of math this problem belongs to?
    somewhere between algebra, analysis and (quantum) physics i guess!
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    Quote Originally Posted by NonCommAlg View Post
    somewhere between algebra, analysis and (quantum) physics i guess!
    I would suggest functional analysis (they are linear operators on sequence spaces).

    CB
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  7. #7
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    Could someone please send me this paper: "Sequence spaces and inverse of an infinite matrix", Bruno De Malafosse, Eberhard Malkowsky. This is the link: SpringerLink - Journal Article
    I believe this paper will help me a lot in studying inverses of infinite matrices, but I don't have access to the full text. I wonder if you could help me download this paper and send it to my mailbox: yemyrina@sina.com.
    Thanks in advance!
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