Eigenvalues and Eigenvectors | Sahithyan's Notes

Let $T: V\rightarrow V$ over $F$ . $\lambda \in F$ and $v \in V$ satisfying $T(v)=\lambda v$ are called an eigenvalue and a corresponding eigenvector of $T$ respectively.

Here for any $\lambda$ , $v=\underline{0}$ is a trivial solution. Usually only non-trivial solutions are mentioned.

Properties

If $\lambda$ is an eigenvalue of $A$ then $\lambda^{-1}$ is an eigenvalue of $A^{-1}$ .

If $A^{-1} = P^{-1} A P$ for some matrix $P$ then $A$ and $A^{-1}$ has the same set of eigenvalues.

Product of the eigenvalues is equal to the determinant.

Sum of the eigenvalues is the equal to the trace of the matrix.

Cayley Hamilton theorem can be used to find the inverses and higher power matrices efficiently.

If $\lambda$ is an eigenvalue of $A$ , then $\lambda^n$ is an eigenvalue of $A^n$ where $n \in \mathbb{Z}$ .

If $\lambda$ is an eigenvalue of $A$ , then $p(\lambda)$ is an eigenvalue of $p(A)$ where $p$ is any polynomial which takes in an number or a matrix.

Eigenspace

For a given $\lambda \in F$ , the set of corresponding eigenvectors.

V_\lambda = \set{ v\in V | T(v)=\lambda v }

An eigenspace is a subspace of $V$ .

Basis of an eigenspace is usually given as the eigenvectors.

For any eigenvalues $\lambda_1$ and $\lambda_2$ ,

V_{\lambda_1} \cap V_{\lambda_2} = \set { \underline{0} }

Geometric Multiplicity

Dimension of $V_\lambda$ . Denoted as $g_\lambda$ .

Minimal Polynomial

For $A \in F^{nn\times n}$ , $m(x)$ of $A$ is such that:

$m(x)$ is monic (i.e. the leading coefficient is 1) (which implies $m(x) \not\equiv 0$ )
$m(A) = 0$
If $h(x) \not\equiv 0$ and $h(A)=0$ then $\text{deg}(h) \ge \text{deg}(m)$

If $f(\lambda)=0$ then $m(\lambda)=0$ .

$m(x)$ divides $f(x)$ .

If $m(\lambda)=0$ then $f(\lambda)=0$ .

As $m(x)$ divides $f(x)$ , they both have the same basic factors with different powers. The powers in $f(x)$ are called algebraic multiplicity ( $a_\lambda$ ). The powers in $m(x)$ are called minimal mulitplicity ( $m_\lambda$ ).

n \ge a_\lambda \ge m_\lambda \ge 1

Diagonalization

Basics for Diagonalization is already covered in 1st semester.

The following statements are equivalent:

$A \in C_{n\times n}$ is diagonalizable
$a_\lambda = g_\lambda$
$m_\lambda = 1$
Eigenvectors form a basis for $C_n$