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 diagonizable. In particular for an $n\times n$ matrix, the matrix is defective if it does not have n linearly independent eigenvectors.

A simple example of a defective matrix is:

$A={\begin{bmatrix}0&1\\0&0\end{bmatrix}}$

Which has a double eigenvalue of 0 but only one eigenvector (1 0) (and multiples thereof).

A defective matrix always has fewer than n distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors.

A Hermitian matrix (or a real symmetric matrix) or a unitary matrix is never defective.

References

• Gilbert Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt: San Diego, 1988).