Linear Algebra Eigenvalues and Diagonalization Cheat Sheet

Key Facts

• A scalar $\lambda$ is an eigenvalue of $A$ if there is a nonzero vector $\mathbf{v}$ such that $A\mathbf{v} = \lambda \mathbf{v}$.
• The characteristic polynomial of an $n \times n$ matrix $A$ is $p(\lambda) = \det(A - \lambda I)$ or equivalently $\det(\lambda I - A)$, depending on convention.
• Eigenvalues are found by solving $\det(A - \lambda I) = 0$.
• The eigenspace for an eigenvalue $\lambda$ is $E_{\lambda} = \operatorname{Null}(A - \lambda I)$.
• An $n \times n$ matrix $A$ is diagonalizable if it has $n$ linearly independent eigenvectors.
• If $A = PDP^{-1}$, then the columns of $P$ are eigenvectors of $A$ and $D$ contains the corresponding eigenvalues on its diagonal.
• If $A = PDP^{-1}$, then $A^k = PD^kP^{-1}$ for any positive integer $k$.
• The algebraic multiplicity of $\lambda$ is its multiplicity as a root of $p(\lambda)$, and the geometric multiplicity is $\dim(E_{\lambda})$.

Vocabulary

Eigenvalue

An eigenvalue $\lambda$ is a scalar for which $A\mathbf{v} = \lambda \mathbf{v}$ has a nonzero solution $\mathbf{v}$.

Eigenvector

An eigenvector is a nonzero vector $\mathbf{v}$ whose direction is unchanged by the transformation $A$, so $A\mathbf{v} = \lambda \mathbf{v}$.

Characteristic Polynomial

The characteristic polynomial is $p(\lambda) = \det(A - \lambda I)$ or $p(\lambda) = \det(\lambda I - A)$, and its roots are the eigenvalues.

Eigenspace

The eigenspace $E_{\lambda}$ is the set of all vectors satisfying $(A - \lambda I)\mathbf{v} = \mathbf{0}$.

Diagonalizable Matrix

A matrix is diagonalizable if it can be written as $A = PDP^{-1}$ for some invertible matrix $P$ and diagonal matrix $D$.

Geometric Multiplicity

Geometric multiplicity is the number of linearly independent eigenvectors associated with an eigenvalue, equal to $\dim(E_{\lambda})$.