Defective eigenvalue

An ${\displaystyle n\times n}$ defective matrix always has fewer than ${\displaystyle n}$ distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues ${\displaystyle \lambda }$ with algebraic multiplicity${\displaystyle m>1}$ (that is, they are multiple roots of the characteristic polynomial), but fewer than ${\displaystyle m}$ linearly independent eigenvectors associated with ${\displaystyle \lambda }$. If the algebraic multiplicity of ${\displaystyle \lambda }$ exceeds its geometric multiplicity (that is, the number of linearly independent eigenvectors associated with ${\displaystyle \lambda }$), then ${\displaystyle \lambda }$ is said to be a defective eigenvalue. However, every eigenvalue with algebraic multiplicity ${\displaystyle m}$ always has ${\displaystyle m}$ linearly independent generalized eigenvectors.