example of non-diagonalizable matrices

Some matrices with real entries which are not diagonalizable over $ℝ$ are diagonalizable over the complex numbers $ℂ$.

For instance,

has ${\lambda }^2+1$ as characteristic polynomial.This polynomial doesn’t factor over the reals, but over $ℂ$ it does. Its roots are $\lambda =\pm i$.

Interpreting the matrix as a linear transformation ${ℂ}^2\to {ℂ}^2$, it has eigenvalues $i$ and $-i$ and linearly independent eigenvectors $(1,-i)$, $(-i,1)$. So we can diagonalize $A$:

But there exist real matrices which aren’t diagonalizable even if complex eigenvectors and eigenvalues are allowed.

For example,

cannot be written as $UD{U}^{-1}$ with $D$ diagonal.

In fact, the characteristic polynomial is ${\lambda }^2$ and it has only one double root $\lambda =0$.However the eigenspace corresponding to the $0$ (kernel) eigenvalue has dimension 1.

$B\left(\begin{array}{c}\hfill v_1\hfill \\ \hfill v_2\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}\right)\iff v_2=0$ and thus the eigenspace is $ker(B)=spa{n}_{ℂ}\left\{{(1,0)}^T\right\}$, with only one dimension.

There isn’t a change of basis where $B$ is diagonal.