Eigen Things

If $A\in {\mathbb{R}}^{n\times n}$, and there is a $\vec{v}\ne 0\in {\mathbb{R}}^n$, and $A\vec{v}=\lambda \vec{v}$ Then $\vec{v}$ is an eigenvectors for A, and $\lambda$ is the eigenvalue.

Eigenspaces

The span of the eigenvector of $A$ is the eigenspace of $A$. It spans a subspace of ${\mathbb{R}}^n$ called the $\lambda$-eigenspace of $A$.

The $\lambda$-eigenspace of $A$ is $\text{Nul}(A-\lambda I)$

$$\begin{array}{rl}A\vec{v}& =\lambda \vec{v}\\ A\vec{v}-\lambda \vec{v}& =0\\ (A-\lambda I)\vec{v}& =0\end{array}$$

Theorems

• The diagonal elements of a triangular matrix are its eigenvalues.
• A not invertible $⟺$ 0 is an eigenvalue of A.
• Stochastic matrices have an eigenvalue equal to 1.
• Eigenvectors with distinct eigenvalues are linearly independent vectors.

Compute Eigenvalues

We know that $(A-\lambda I)$ is non invertible, so $\det (A-\lambda I)=0$. We can solve $\det (A-\lambda I)=0$ for $\lambda$.

$\det (A-\lambda I)$ is the characteristic polynomial $\det (A-\lambda I)=0$ is the characteristic equation The trace of a matrix is the sum of its diagonal elements. The sum of the Eigenvalues of $A$ = The trace.

Algebraic and Geometric Multiplicities

• $a_i$ is the algebraic multiplicity
• $g_i$ is the geometric multiplicity
• $1\le a_i\le n$
• $1\le g_i\le n$
• $g_i\le a_i$