I have a matrix
M with real entries in
MachinePrecision. I compute its eigenvectors and construct matrix
ev = Eigenvectors[M]; U = Transpose @ ev;
U is supposed to be an orthogonal matrix and thus
Transpose[U].U == IdentityMatrix[ Length @ U].
Mathematica gives the latter equality with the accuracy of
Transpose[U].U - IdentityMatrix[ Length @ U]// Abs// Max == 8.8131 * 10^(-16).
Is it possible to make this difference smaller, let's say, order of
10^(-32)? I need to get eigenvectors which are orthonormal with accuracy of order of
10(-32) or higher.