Defective matrix

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a defective matrix is a <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> that does not have a complete <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a> of <a href="/facts/Eigenvector/8TjEoT8u">eigenvectors</a>, and is therefore not <a href="/facts/Diagonalizable_matrix/y3MkyTCy">diagonalizable</a>. In particular, an 
 
 
 
 n
 ×
 n
 
 
 {\displaystyle n\times n}
 
 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> is defective <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> it does not have 
 
 
 
 n
 
 
 {\displaystyle n}
 
 <a href="/facts/Linearly_independent/8JrHfIIa">linearly independent</a> eigenvectors. A complete basis is formed by augmenting the eigenvectors with <a href="/facts/Generalized_eigenvector/Lf7MgEGJ">generalized eigenvectors</a>, which are necessary for solving defective systems of <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equations</a> and other problems.
An 
 
 
 
 n
 ×
 n
 
 
 {\displaystyle n\times n}
 
 defective matrix always has fewer than 
 
 
 
 n
 
 
 {\displaystyle n}
 
 distinct <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a>, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 with <a href="/facts/Algebraic_multiplicity/8TjEoT8u">algebraic multiplicity</a> 
 
 
 
 m
 >
 1
 
 
 {\displaystyle m>1}
 
 (that is, they are multiple <a href="/facts/Root_of_a_polynomial/mlK3dhoT">roots</a> of the <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a>), but fewer than 
 
 
 
 m
 
 
 {\displaystyle m}
 
 linearly independent eigenvectors associated with 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
. If the algebraic multiplicity of 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 exceeds its <a href="/facts/Geometric_multiplicity/8TjEoT8u">geometric multiplicity</a> (that is, the number of linearly independent eigenvectors associated with 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
), then 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 is said to be a defective eigenvalue. However, every eigenvalue with algebraic multiplicity 
 
 
 
 m
 
 
 {\displaystyle m}
 
 always has 
 
 
 
 m
 
 
 {\displaystyle m}
 
 linearly independent generalized eigenvectors.
A <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric matrix</a> and more generally a <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian matrix</a>, and a <a href="/facts/Unitary_matrix/8fsxv3KD">unitary matrix</a>, is never defective; more generally, a <a href="/facts/Normal_matrix/WVKxrEK9">normal matrix</a> (which includes Hermitian and unitary matrices as special cases) is never defective.

Defective matrix open-in-new

Defective matrix