Eigenvalue Question: Can I have an Infinite Matrix?

[cragar]

Can I have a matrix that has an uncountable number of eigenvalues?
If the matrix was infinite.
And also can I have a matrix with a countable number of rows and an uncountable number of
columns?

 

[micromass]

Sure, you can have that. But we often don't speak of "infinite dimensional matrix" anymore, but rather of a linear operator.

 

[cragar]

Ok thanks for your answer. What about my second question?
Can I have a matrix with a countable number of rows and an uncountable number of
columns?

 

[micromass]

The yes was to both questions.

 

[cragar]

ok thanks. Are there any other crazy interesting properties of infinite matrices?

 

[micromass]

The craziest property, I think, is that infinite matrices don't need to be continuous. This is quite a serious defect, since discontinuous linear maps are not so interesting.

 

[algebrat]

a matrix with countably many rows and uncountably many columns might be a linear map from functions to sequences.