Find Eigenvalues/Determinant of Infinite Matrix

[cragar]

If I had an infinite matrix $\aleph_0 \times \aleph_0$ could I find the eigenvalues or the Determinant of this matrix. I think some of these matrices would have a finite Determinant or it could be zero. Because i could add 1/2+1/4+1/8... but I would just need a matrix with the right entries. Just wondering if anyone has done this and how you would go about figuring it out.

 

[chiro]

Hey cragar.

The topic that deals with this kind of thing is the Hilbert-Space theory that deals with operator algebras in infinite-dimensional spaces.

If you want to look into this look into things like Hilbert-Space Theory, Banach Spaces, and operator algebras like C* algebras as well as functional analysis in the infinite-dimensional spaces.

 

[HallsofIvy]

Note that you will need to have some kind of "regularity conditions" on the "infinite matrices" in order that the infinite sums involved will converge.

 

[Byron Chen]

Hey cragar.

The topic that deals with this kind of thing is the Hilbert-Space theory that deals with operator algebras in infinite-dimensional spaces.

If you want to look into this look into things like Hilbert-Space Theory, Banach Spaces, and operator algebras like C* algebras as well as functional analysis in the infinite-dimensional spaces.

Can you recommend any sources related to these topics? Thanks.