Defective matrix

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 ${\displaystyle n\times n}$ matrix is defective if and only if it does not have ${\displaystyle n}$ 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 ${\displaystyle n\times n}$ defective matrix always has fewer than ${\displaystyle n}$ distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues ${\displaystyle \lambda }$ with algebraic multiplicity ${\displaystyle m>1}$ (that is, they are multiple roots of the characteristic polynomial), but fewer than ${\displaystyle m}$ linearly independent eigenvectors associated with ${\displaystyle \lambda }$. If the algebraic multiplicity of ${\displaystyle \lambda }$ exceeds its geometric multiplicity (that is, the number of linearly independent eigenvectors associated with ${\displaystyle \lambda }$), then ${\displaystyle \lambda }$ is said to be a defective eigenvalue.^[1] However, every eigenvalue with algebraic multiplicity ${\displaystyle m}$ always has ${\displaystyle m}$ 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.