Intuitive conditions to recognize a non diagonalizable matrix.

Question

I know that to find out whether or not a matrix is diagonalizable you have to study eigenvectors and eigenvalues and the space they generate. I also know some shortcuts to find out if it is diagonalizable (Spectral theorem, study the ker before approaching the Characteristic polynomial...). But what i'd like to ask you is if there is any method or intuition to find out almost instantly, or with few calculations, if the matrix is non diagonalizable and what criteria would you use to write a simple one.

Thanks to everyone reading! :D

Answer 1

This is difficult even for $2\times2$ matrices. One complication is that diagonalisability is a field-dependent property. E.g. every real $2\times2$ rotation matrix $R\ne\pm I$ is diagonalisable over $\mathbb C$ but not over $\mathbb R$. When the field is not algebraically closed, you really need to see whether the characteristic equation is solvable first.

However, if the field is algebraically closed, it is easy to spot a non-diagonalisable $2\times2$ matrix $A$ quickly: $A$ is non-diagonalisable if and only if it is not a scalar matrix and $\operatorname{tr}(A)^2=4\det(A)$.

For larger-sized matrices, I don't think there is any easy trick to determine whether a generic matrix is diagonalisable or not.

Answer 2

I think that it is highly unlikely (to say the least) that someone will develop enough intuition to tell you, without computations, that the matrix\begin{bmatrix}-7 a-56 & -7 a-35 & 7-21 a \\ 11 a+88 & 11 a+55 & 33 a-11 \\ -a-8 & -a-5 & 1-3 a\end{bmatrix}is diagonalizabele when and only when $a\neq0$.