# Eigenvalue question

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?

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

Can I have a matrix with a countable number of rows and an uncountable number of
columns?

The yes was to both questions.

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

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.

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