# What guarantees that joint diagonalization is possible?

According to the Wikipedia article on the common spatial pattern algorithm, one can find the following matrices by joint diagonalization of a pair of covariance matrices $R_1$ and $R_2$:

$$P = [\mathbf{p_1}, \cdots , \mathbf{p_n}]$$ $$D = \mbox{diag} \{\lambda_1, \cdots ,\lambda_n\}$$

such that $\mathbf{P^{-1}R_1P=D}$ and $\mathbf{P^{-1}R_2P=I_n}$, where $\mathbf{I_n}$ is the identity matrix of rank $n$.

Now, I know that there are "plenty" of non-diagonalizable matrices, so I'm inclined to wonder what guarantees the above is possible, and when.

-
I read the article and I'm baffled too... Isn't $P^{-1}R_2P = I$ the same as $R_2 = I$? – Tunococ May 27 '13 at 2:39
Then I'd say it is jointly diagonalizable if $R_2 = I$, and you only need to diagonalize $R_1$ because no matter what $P$ and $D$ are, $P^{-1}R_2P = I$ is guaranteed. I feel that the Wikipedia page you're referring to is not very reliable. – Tunococ May 27 '13 at 2:48
If $P^{-1}R_2P$ is diagonal, it must contain eigenvalues of $R_2$, so you at least require $R_2$ to have only one distinct eigenvalue. – Tunococ May 27 '13 at 5:46
I concur with Tunococ. The Wikipedia article you cited does not sound right. If $P^{-1}R_2P=I_n$, then $R_2$ is necessarily equal to $I_n$, but by assumption, $R_2$ is just an arbitrary covariance matrix. In general, it is not the identity matrix. – user1551 May 27 '13 at 10:39
In general, two square matrices $\,A,B\,$ are simultaneously diagonalizable (which seems to be more or less the same as jointly diagonalizable) iff there are diagonalizable and $\,AB=BA\,$ ... – DonAntonio May 27 '13 at 21:25