mateigen

Function: mateigen
Section: linear_algebra
C-Name: mateigen
Prototype: GD0,L,p
Help: mateigen(x,{flag=0}): complex eigenvectors of the matrix x given as
 columns of a matrix H. If flag=1, return [L,H], where L contains the
 eigenvalues and H the corresponding eigenvectors.
Doc: returns the (complex) eigenvectors of $x$ as columns of a matrix.
 If $\fl=1$, return $[L,H]$, where $L$ contains the
 eigenvalues and $H$ the corresponding eigenvectors; multiple eigenvalues are
 repeated according to the eigenspace dimension (which may be less
 than the eigenvalue multiplicity in the characteristic polynomial).

 This function first computes the characteristic polynomial of $x$ and
 approximates its complex roots $(\lambda_i)$, then tries to compute the
 eigenspaces as kernels of the $x - \lambda_i$. This algorithm is
 ill-conditioned and is likely to miss kernel vectors if some roots of the
 characteristic polynomial are close, in particular if it has multiple roots.
 \bprog
 ? A = [13,2; 10,14]; mateigen(A)
 %1 =
 [-1/2 2/5]

 [   1   1]
 ? [L,H] = mateigen(A, 1);
 ? L
 %3 = [9, 18]
 ? H
 %4 =
 [-1/2 2/5]

 [   1   1]
 ? A * H == H * matdiagonal(L)
 %5 = 1
 @eprog\noindent
 For symmetric matrices, use \tet{qfjacobi} instead; for Hermitian matrices,
 compute
 \bprog
  A = real(x);
  B = imag(x);
  y = matconcat([A, -B; B, A]);
 @eprog\noindent and apply \kbd{qfjacobi} to $y$.
Variant: Also available is \fun{GEN}{eigen}{GEN x, long prec} ($\fl = 0$)