Diagonalize a symmetric matrix in Maxima

Question

Here is my problem, I have a real symmetric matrix M depending on two parameters a,b (that are assumed to be positive) and I want to find an orthogonal matrix P such that PMP^{-1} is a diagonal matrix. Here is an example of what I've done :

assume(a>0,b>0);
M : matrix([a,a+b,a+b],[a+b,a,a+b],[a+b,a+b,a]);
load("eigen");
[myeigval,myeigvec]:similaritytransform(ev(M,hermitianmatrix));

or simply,

assume(a>0,b>0);
M : matrix([a,a+b,a+b],[a+b,a,a+b],[a+b,a+b,a]);
load("eigen");
[myeigval,myeigvec]:similaritytransform(M);

I get the same result for this two tests :

[[[2*b+3*a,-b],[1,2]],[[[1/sqrt(3),1/sqrt(3),1/sqrt(3)]],[[1/sqrt(2),0,-1/sqrt(2)],[0,1/sqrt(2),-1/sqrt(2)]]]]

The norm of the vectors is equal to 1 but this vectors do not give me an orthogonal matrix. Can somebody explain to me what is the problem?

Answer 1

Look at the global variables rightmatrix and leftmatrix after you call similaritytransform. When I try your example, I find that rightmatrix . leftmatrix is the identity matrix (and so is leftmatrix . rightmatrix).

I agree the documentation for similaritytransform isn't entirely clear. Oh well.