Re:Clojure Linear Algebra Refresher: Eigenvalues and Eigenve...
by neutronicus on Tuesday June 06, 2017 @01:32PM
(#14499143)
The dimension of the space spanned by the eigenvectors associated with a degenerate eigenvalue is bounded above by the algebraic multiplicity of the eigenvalue and bounded below by one.
Re:Clojure Linear Algebra Refresher: Eigenvalues and Eigenve...
by btilly on Tuesday June 06, 2017 @01:28PM
(#14499101)
Not true. The matrix [[1 1] [0 1]] has the eigenvalue 1 repeated twice but the eigenvectors are on a line.
The generalized eigenvectors are on a plane.
The case where the eigenvectors do not span the space corresponds to matrices whose Jordan normal form has upper triangular bits.
[edit: the rest of my original comment was not correct]
If an eigenvalue is unique, the corresponding eigenvectors are on a line. If it appears twice, the corresponding eigenvectors are [edit: could be] on a plane. In general, they span [edit: could span] a subspace of dimension equal to the multiplicity.