Defective matrix
Non-diagonalizable matrix; one lacking a basis of eigenvectors / From Wikipedia, the free encyclopedia
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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, an matrix is defective if and only if it does not have
linearly independent eigenvectors.[1] 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.
An defective matrix always has fewer than
distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues
with algebraic multiplicity
(that is, they are multiple roots of the characteristic polynomial), but fewer than
linearly independent eigenvectors associated with
. If the algebraic multiplicity of
exceeds its geometric multiplicity (that is, the number of linearly independent eigenvectors associated with
), then
is said to be a defective eigenvalue.[1] However, every eigenvalue with algebraic multiplicity
always has
linearly independent generalized eigenvectors.
A real symmetric matrix and more generally a Hermitian matrix, and a unitary matrix, is never defective; more generally, a normal matrix (which includes Hermitian and unitary matrices as special cases) is never defective.