Repeated eigenvalues

1. Apr 17, 2012

LikeMath

Hi there!

Let A be a square matrix of order n.
It is well known that if we have n distinct eigenvalues then we surely have n distinct eigenvectors. But if there are repeated eigenvalues then the tow possibilities may happen.
My question is: How can I know that do the eigenvectors are distinct or not?

Thank you very much.

2. Apr 17, 2012

HallsofIvy

The only way to be certain is by trying to find the eigenvectors!

For example both
$$\begin{bmatrix}a & 0 \\ 0 & a\end{bmatrix}$$
and
$$\begin{bmatrix} a & 1 \\ 0 & a\end{bmatrix}$$
have a as a double eigenvalue.
To find the eigenvectors, we need to solve
$$\begin{bmatrix}a & 0 \\ 0 & a\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}ax \\ ay\end{bmatrix}$$
and
$$\begin{bmatrix}a & 1 \\ 0 & a\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}ax \\ ay\end{bmatrix}$$

The first gives the two equations ax= ax and ay= ay. Clearly, those are true for all x and y- any vector in R2 and, in particular <1, 0> and <0 1>, which are independent, are eigenvectors.

The second gives the two equations ax+ y= ax and ay= ay. The second equation is true for any y but the first equation reduces to y= 0. Given that a can be anything but we have that all eigenvectors are of the form <a, 0>, a one dimensional space so we have only one "independent" eigenvector.