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Theorem.

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## Homework Statement

Let A be a square matrix and [tex]f_A (\lambda)[/tex] its characteristic polynomial. In each of the following cases (i) to (iv), write down whether

A is diagonalizable over R

A is not diagonalizable over R

its not possible to say one way or the other.

THEN say whether or not the matrix can certainly be diagonalizable over C

I am only having trouble with the following case:

[tex]f_A (\lambda)=(\lambda^2+3)^2[/tex]

## Homework Equations

There aren't any relevant equations strictly speaking

## The Attempt at a Solution

Well it is clear that the eigenvalues are [tex]\sqrt{3}i and -\sqrt{3}i [/tex], both having multiplicity 2 in the characteristic polynomial. Clearly then A is not diagonalizable over R as its eigenvalues are not real. Where I get stuck is deciding if the matrix can certainly be diagonalizable over C or not. from the characteristic polynomial I see that A is 4x4, and it does not have 4 distinct eigenvalues, which doesn't help me. I am not sure how to approach this really its the first time I have encoutered a characteristic polynomial of this sort and cant seem to find anything helpful in my textbook. Any help/advice would be greatly appreciated

-Curtis