defective matrix
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English
Noun
defective matrix (plural defective matrices)
- (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 Lid.
- 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,
Usage notes
Equivalently, a matrix is defective if one of its eigenvalues has geometric multiplicity less than its algebraic multiplicity.
See also
Further reading
Basis (linear algebra) on Wikipedia.Wikipedia
