I think you want a text like Hoffman-Kunze which gives the basic definitions and theorems over an arbitrary ground field. At some point the field starts to matt

I think you want a text like Hoffman-Kunze which gives the basic definitions and theorems over an arbitrary ground field. At some point the field starts to matter -- giving the real plane a quarter turn has no real eigenvalues, for example; and I don't see a good way to transport the statement of SVD to the setting of finite fields.