Generalized Eigenvector
In linear algebra, for a matrix A, there may not always exist a full set of linearly independent eigenvectors that form a complete basis – a matrix may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue λ is greater than its geometric multiplicity (the nullity of the matrix, or the dimension of its nullspace). In such cases, a generalized eigenvector of A is a nonzero vector v, which is associated with λ having algebraic multiplicity k ≥1, satisfying
The set of all generalized eigenvectors for a given λ, together with the zero vector, form the generalized eigenspace for λ.
Ordinary eigenvectors and eigenspaces are obtained for k=1.
Read more about Generalized Eigenvector: For Defective Matrices, Other Meanings of The Term
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