# Defective matrix

From Encyclopedia of Mathematics

A matrix $A\in\mathbf C^{n\times n}$ is called non-defective if it has a set of $n$ independent eigenvectors (cf. Eigen vector). Otherwise it is called defective. The notion is of particular importance in numerical linear algebra.

#### Comment

A complex matrix $A$ is non-defective if and only if it is similar to a diagonal matrix: $\Delta = P A P^{-1}$.

#### References

- D.M. Young, R.T. Gregory, "A survey of numerical mathematics" ,
**2**, Dover, reprint (1988) pp. 741–743 - Lloyd N. Trefethen, David Bau III,
*Numerical Linear Algebra*SIAM (1997)**ISBN**0898713617

**How to Cite This Entry:**

Defective matrix.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Defective_matrix&oldid=54544

This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article