Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
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

1 Answer 1

The most likely explanation is that they are referring to joint approximate diagonalization by an orthogonal matrix the minimizes the sum of the Frobenius norm of the off-diagonal terms. The algorithm used is often called JADE and a quick web search for JADE and common spatial pattern picks up many promising hits.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.