I want to diagonalize a matrix and then be able to do basis changes. The aim in the end is to do matrix exponentiation, with exp(A) = P.exp(D).P^{-1}.
I use sgeev to diagonalize A. If I am not mistaken (and I probably am since it's not working), sgeev gives me P in the vr matrix and P^{-1} is transpose(vl). The diagonal matrix can be reconstitute from the eigenvalues wr.
The problem is that when I try to verify the matrix transformation by computing P * D * P^{-1} it's not giving A back.
Here's my code:
integer :: i,n, info
real::norm
real, allocatable:: A(:,:), B(:,:), C(:,:),D(:,:)
real, allocatable:: wr(:), wi(:), vl(:, :), vr(:, :), work(:)
n=3
allocate(vr(n,n), vl(n,n), wr(n), wi(n), work(4*n))
allocate(A(n,n),B(n,n), C(n,n),D(n,n))
A(1,:)=(/1,0,1/)
A(2,:)=(/0,2,1/)
A(3,:)=(/0,3,1/)
call sgeev('V','V',n,A,n,wr,wi,vl,n,vr,n,work,size(work,1),info)
print*,'eigenvalues'
do i=1,n
print*,i,wr(i),wi(i)
enddo
D=0.0
D(1,1)=wr(1)
D(2,2)=wr(2)
D(3,3)=wr(3)
C = matmul(D,transpose(vl))
B = matmul(vr,C)
print*,'A'
do i=1, n
print*, B(i,:)
enddo
The printed result is:
eigenvalues
1 1.00000000 0.00000000
2 3.30277562 0.00000000
3 -0.302775621 0.00000000
A
0.688247263 0.160159975 0.764021933
0.00000000 1.66581571 0.817408621
0.00000000 2.45222616 0.848407149
A is not the original A, not even considering an eventual factor.
I guess I am somehow mistaken since I checked the eigenvectors by computing matmul(A,vr) = matmul(vr,D) and matmul(transpose(vl),A) = matmul(D, transpose(vl)), and it worked.
Where am I wrong?
The problem is that transpose(vl) is not the inverse of vr. The normalisation given by sgeev is that each eigenvector (each column of vl or vr) is individually normalised. This means that dot_product(vl(:,i), vr(:,j)) is zero if i/=j, but is in general <1 if i==j.
If you want to get P^{-1}, you need to scale each column of vl by a factor of 1/dot_product(vl(:,i),vr(:,i) before transposing it.