(

linear algebra,

analysis:)

An *n* x *n* matrix *A* is called **defective** if it has less than *n* linearly independent eigenvectors. This means that one cannot construct an eigenbasis for *A*, or a basis for **R**^{n} contructed out of eigenvectors for *A*.

One method of determining if *A* is defective is to determine the characteristic polynomial of *A*, p_{A}, noting that each root is an eigenvalue, and that for each eigenvalue *c*, if (x - c)^{k} divides p_{A} and *k* > geometric multiplicity, then *A* is defective. In such a case, *c* is said to have algebraic multiplicity *k*.