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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 ''n'' × ''n'' 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 associated with ''λ''. If the algebraic multiplicity of ''λ'' exceeds its geometric multiplicity, then ''λ'' is said to be a defective eigenvalue.〔 However, every eigenvalue with algebraic multiplicity ''m'' always 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; more generally, a normal matrix (which includes Hermitian and unitary as special cases) is never defective. == Jordan block == Any Jordan block of size 2×2 or larger is defective. For example, the n × n Jordan block, : has an eigenvalue, λ, with algebraic multiplicity n, but only one distinct eigenvector, : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「defective matrix」の詳細全文を読む スポンサード リンク
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