**For Defective Matrices**

Generalized eigenvectors are needed to form a complete basis of a defective matrix, which is a matrix in which there are fewer linearly independent eigenvectors than eigenvalues (counting multiplicity). Over an algebraically closed field, the generalized eigenvectors *do* allow choosing a complete basis, as follows from the Jordan form of a matrix.

In particular, suppose that an eigenvalue *λ* of a matrix *A* has an algebraic multiplicity *m* but fewer corresponding eigenvectors. We form a sequence of *m* eigenvectors and generalized eigenvectors that are linearly independent and satisfy

for some coefficients, for . It follows that

The vectors can always be chosen, but are not uniquely determined by the above relations. If the geometric multiplicity (dimension of the eigenspace) of *λ* is *p*, one can choose the first *p* vectors to be eigenvectors, but the remaining *m* − *p* vectors are only generalized eigenvectors.

Read more about this topic: Generalized Eigenvector

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