defective matrix

defective matrix (plural defective matrices)

1. (linear algebra) A square (n×n) matrix that has fewer than n linearly independent eigenvectors, and is therefore not diagonalisable.
   • 1994, Sigal Ar, Jin-Yi Cai, Chapter 5: Reliable Benchmarks Using Numerical Stability, Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, Association for Computing Machinery, Society for Industrial and Applied Mathematics, page 41,

     An n x n matrix A is a defective matrix if it has a defective eigenvalue.

     Obviously, a Jordan block of dimension greater than 1, and a matrix whose Jordan canonical form has a Jordan block of dimension greater than 1, are defective matrices.

   • 2007, Thomas S. Shores, Applied Linear Algebra and Matrix Analysis, Springer, page 261:

     Therefore, the sum of the geometric multiplicities of a defective matrix will be less than n.

   • 2013, Angelo Luongo, “Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures”, in Alexander F. Vakakis, editor, Normal Modes and Localization in Nonlinear Systems, Springer, page 153:

     Thus, defective matrices exhibit high sensitivity to imperfections.
     It can be checked that the unperturbed matrices L* are in fact nearly-defective, because they have nearly-parallel eigenvectors. In particular, the matrices are themselves perturbations of order ε of exactly defective matrices L_id.

Equivalently, a matrix is defective if one of its eigenvalues has geometric multiplicity less than its algebraic multiplicity.

• characteristic polynomial
• defective eigenvalue

• Basis (linear algebra) on Wikipedia.Wikipedia