Eigenvalue question

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.