Dear Maple experts,
I am struggling with a difference between the symbolic and numerical solution of an eigendecomposition of a symmetric positive definite matrix. Numerically the solution seems correct, but the symbolic solution puzzles me. In the symbolic solution the reconstructed matrix is different from the original matrix (although the difference between the original and the reconstructed matrix seems to be related to an unknown scalar multiplier.
Lambda := Matrix(5, 1, symbol = lambda);
Theta := Matrix(5, 5, shape = diagonal, symbol = theta);
#Ω is the matrix that will be diagonalized.
Omega := MatrixPower(Theta, -1/2) . Lambda . Lambda^%T . MatrixPower(Theta, -1/2);
#Ω is symmetric and in practice always positive definite, but I do not know how to specify the assumption of positivess definiteness in Maple
# the matrix Omega is very simple and Maple finds a symbolic solution
E, V := Eigenvectors(Omega);
# this will not return the original matrix
simplify(V . DiagonalMatrix(E) . V^%T)
# check this numerically with the following values.
lambda[1, 1] := .9;lambda[2, 1] := .8;lambda[3, 1] := .7;lambda[4, 1] := .85;lambda[5, 1] := .7;
theta[1, 1] := .25;theta[2, 2] := .21;theta[3, 3] := .20;theta[4, 4] := .15;theta[5, 5] := .35;
The dotproduct is not always zero, although I thought that the eigenvectors should be orthogonal.
I know eigenvector solutions may be different because of scalar multiples, but here I am not able to understand the differences between the numerical and symbolic solution.
I probably missed something, but I spend the whole saturday trying to solve this problem, but I can not find it.
I attached both files.
Anyone? Thank in advance,