Do you know if eigenvectors of idempotent matrices are always orthonormal?

Question

I'm currently studying Econometrics theory, and I'm stuck in my problem set where it is asked if "eigenvectors of idempotent matrices are always orthonormal". How can I justify that?

Cheers.

Answer 1

The eigenvectors need not be orthogonal. If you like to think of it geometrically, consider an obligue projection.

As an example, consider: $$\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}$$

You can check for yourself that it is idempotent. Since it is upper triangular, we can read off its eigenvalues on the main diagonal; they are $1$ and $0$.

$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ is an eigenvector for the eigenvalue $1$. And $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$ is an eigenvector for the eigenvalue $0$. The angle between these vectors is $\frac{\pi}{4}$ (or $45^\circ$). You can norm the latter eigenvector so it obtains unit length, if you like, but you cannot change the fact that the eigenspaces are not perpendicular. (Orthonormal means both orthogonal and normed.)

You can think of the matrix $\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}$ as an oblique projection that projects along the "skew" direction given by the second eigenvector onto the first coordinate axis (the subspace spanned by the first eigenvector).

Answer 2

No, and it's easy to construct an example where they are not. If $P$ is any invertible matrix, and $D$ is a diagonal matrix with $0$ and $1$ on the diagonal, then $$ PDP^{-1} $$ is just as idempotent as $D$ is, and its eigenvectors are the columns of $P$. You're free to choose the columns of $P$ to be non-orthogonal.