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