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In linear algebra, a 'defective matrix' is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, for an $n\; imes\; n$ matrix, the matrix is defective if (and only if) it does not have ''n'' linearly independent eigenvectors. A complete basis is formed by augmenting the eigenvectors with generalized eigenvectors, which are necessary for solving defective systems of ordinary differential equations and other problems.
A defective matrix always has fewer than ''n'' distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues λ with algebraic multiplicity $m\; >\; 1$ (that is, they are multiple roots of the characteristic polynomial), but fewer than ''m'' linearly independent eigenvectors. However, every eigenvalue with multiplicity ''m'' has ''m'' linearly independent generalized eigenvectors.
A Hermitian matrix (or the special case of a real symmetric matrix) or a unitary matrix is never defective.

Contents

Example

References

Example

A simple example of a defective matrix is:
:
which has a double eigenvalue of 0 but only one eigenvector
:
(and constant multiples thereof).

References

★ Gilbert Strang, ''Linear Algebra and Its Applications'', 3rd ed. (Harcourt: San Diego, 1988).