Prove a matrix is not diagonalizable

Question

To show that a matrix is not diagonalizable, I would just have to show that there are no eigenvalues present in the matrix.

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So, for example, if I want to prove that

$$A=\begin{bmatrix} 0 & -1 \\1 & 0 \end{bmatrix}$$

is not diagonalizable - would I say that it is since the tr(A) does not form any eigenvalues?

Answer 1

You can compute the charactersitic polynomial, which is in this case equal to $x^{2}+1$. Assuming you are working in $\mathbf{R}$ this polynomial has no real roots, and hence A has no eigenvalues so A is not diagonalisable.

Answer 2

If A (2x2) is diagonalizable, then there must be enough eigenspaces of total size dim(R^2)=2, in other words the sum of the total vectors in all eigenspaces must span R^2.

So in your case, solve for the eigenvalues then their respective eigenspaces. If you get enough a total of 2 vectors across all the eigenspaces, then A is diagonalizable.