How to calculate the inverse of an infinite matrix?

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October 4th 2009, 02:08 PM
zzzhhh

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!
October 4th 2009, 05:47 PM
Jameson

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.
October 4th 2009, 05:51 PM
NonCommAlg

Quote:

Originally Posted by zzzhhh

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.
October 5th 2009, 02:58 AM
zzzhhh

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?
October 5th 2009, 03:21 AM
NonCommAlg

Quote:

Originally Posted by zzzhhh

Could you tell me which branch of math this problem belongs to?

somewhere between algebra, analysis and (quantum) physics i guess!
October 5th 2009, 04:34 AM
CaptainBlack

Quote:

Originally Posted by NonCommAlg

somewhere between algebra, analysis and (quantum) physics i guess!

I would suggest functional analysis (they are linear operators on sequence spaces).

CB
October 7th 2009, 05:57 AM
zzzhhh

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!