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Massimo Zanetti
on 12 Oct 2016

[V,e]=eig(S1,S2) is not "simultaneous diagonalization" of S1,S1 in the sense that they have the same eigenvectors. They DO NOT have the same eigenvectors. It is called "simoultaneous diagonalization" because it gives the right-eigenvectors of both A and (after a simple manipulation) the ones of B.

In fact, V is the matrix of right-eigenvectors of A, and (D^-1*A)*V is the set of right-eigenvectors of B.

Massimo Zanetti
on 12 Oct 2016

Edited: Massimo Zanetti
on 12 Oct 2016

They are not captured by the syntax of eig(A,B). Let me explain you why:

The call

[V,D] = eig(A,B)

returns diagonal matrix D of generalized eigenvalues and full matrix V whose columns are the corresponding right eigenvectors of A, so that A*V = B*V*D.

Now, from this you easily get the right eigenvectors of B (that are not the same to that of A) B*V=D^-1*A*V. Assume that there exists a common eigenvector w for A and B, then Aw=Bw=h. Applying the equivalence above you get Bw*D = D^-1*Aw, and thus DhD=h. This is true only if D is the identity matrix.

yabes dwi nugroho
on 2 Dec 2016

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