# example of non-diagonalizable matrices

Some matrices with real entries which are not diagonalizable over $\mathbb{R}$ *are* diagonalizable over the complex numbers^{} $\u2102$.

For instance,

$$A=\left(\begin{array}{cc}\hfill 0\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$$ |

has ${\lambda}^{2}+1$ as characteristic polynomial^{}.
This polynomial^{} doesn’t factor over the reals, but over $\u2102$ it does. Its roots are $\lambda =\pm i$.

Interpreting the matrix as a linear transformation ${\u2102}^{2}\to {\u2102}^{2}$, it has eigenvalues^{} $i$ and $-i$ and linearly independent^{} eigenvectors^{} $(1,-i)$, $(-i,1)$. So we can diagonalize $A$:

$$A=\left(\begin{array}{cc}\hfill 0\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill 1\hfill & \hfill -i\hfill \\ \hfill -i\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{cc}\hfill i\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -i\hfill \end{array}\right)\left(\begin{array}{cc}\hfill .5\hfill & \hfill .5i\hfill \\ \hfill .5i\hfill & \hfill .5\hfill \end{array}\right)$$ |

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

For example,

$$B=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$$ |

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 $ker(B)=spa{n}_{\u2102}\left\{{(1,0)}^{T}\right\}$, with only one dimension.

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

Title | example of non-diagonalizable matrices |
---|---|

Canonical name | ExampleOfNondiagonalizableMatrices |

Date of creation | 2013-03-22 14:14:30 |

Last modified on | 2013-03-22 14:14:30 |

Owner | cvalente (11260) |

Last modified by | cvalente (11260) |

Numerical id | 14 |

Author | cvalente (11260) |

Entry type | Example |

Classification | msc 15-00 |