# multiplicity of eigenvalues

(1 hours to learn)

## Summary

The characteristic polynomial of a matrix may have repeated roots. If this is the case, the geometric multiplicity of a given eigenvalue (the dimension of the corresponding eigenspace) may be less than the algebraic multiplicity. This property determines whether a matrix is diagonalizable, and it is relevant to the solutions of differential equations.

## Context

This concept has the prerequisites:

- eigenvalues and eigenvectors
- roots of polynomials (The Fundamental Theorem of Algebra implies the sum of the algebraic multiplicities is the dimension of the space.)

## Core resources (read/watch one of the following)

## -Free-

→ A First Course in Linear Algebra (2012)

A linear algebra textbook with proofs.

Location:
Section "Properties of eigenvalues and eigenvectors," subsection "Multiplicities of eigenvalues"

## -Paid-

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location:
Section 9.2, "Eigenvalues, eigenvectors, and applications," subsection 9.2.2, "Diagonalizability," pages 429-433

## See also

-No Additional Notes-