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