Eigenvalues and Eigenvectors

linear transformations

Let $𝝋 : 𝑼 \to 𝑼$ be a linear transformation. A non-zero vector $\vec{𝒖}$ is an eigenvector for $𝝋$ if

𝝋 (\vec{𝒖}) = λ \vec{𝒖}

The value $λ$ is eigenvalue for the eigenvector $\vec{𝒖}$ .

Let $M$ be an $n \times n$ matrix. A non-zero vector $\vec{𝒖}$ is an eigenvector for $M$ if

M \vec{𝒖} = λ \vec{𝒖}

The value $λ$ is eigenvalue for the eigenvector $\vec{𝒖}$ .

Let $L$ be an $n \times n$ matrix.

\begin{matrix} L^{0} & = & I_{n} \\ L^{1} & = & L I_{n} \\ L^{2} & = & L L^{1} \\ L^{3} & = & L L^{2} \\ ⋮ \\ L^{k + 1} & = & L L^{k} \end{matrix}

For a polynomial

\begin{matrix} p (x) & = & a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{k - 1} x^{k - 1} + x^{k} \\ = & a_{0} x^{0} + a_{1} x^{1} + a_{2} x^{2} + \dots + a_{k - 1} x^{k - 1} + x^{k} \\ = & {(x - λ_{1})}^{m_{1}} {(x - λ_{2})}^{m_{2}} \dots {(x - λ_{t})}^{m_{t}} \end{matrix}

and $n \times n$ matrix $L$ define

\begin{matrix} p (L) & = & a_{0} L^{0} + a_{1} L^{1} + a_{2} L^{2} + \dots + a_{k - 1} L^{k - 1} + L^{k} \\ = & {(L - λ_{1} I_{n})}^{m_{1}} {(L - λ_{2} I_{n})}^{m_{2}} \dots {(L - λ_{t} I_{n})}^{m_{t}} \end{matrix}

For a non-zero vector $\vec{𝒖}$ and matrix $L$

\begin{matrix} {\vec{𝒖}}_{0} & = & \vec{𝒖} & = & L^{0} \vec{𝒖} \\ {\vec{𝒖}}_{1} & = & L {\vec{𝒖}}_{0} & = & L^{1} \vec{𝒖} \\ {\vec{𝒖}}_{2} & = & L {\vec{𝒖}}_{1} & = & L^{2} \vec{𝒖} \\ ⋮ \\ {\vec{𝒖}}_{k + 1} & = & L {\vec{𝒖}}_{k} & = & L^{k + 1} \vec{𝒖} \\ ⋮ \end{matrix}

\begin{matrix} \vec{𝒐} & = & α_{0} {\vec{𝒖}}_{0} \\ \vec{𝒐} & = & α_{0} {\vec{𝒖}}_{0} + α_{1} {\vec{𝒖}}_{1} \\ \vec{𝒐} & = & α_{0} {\vec{𝒖}}_{0} + α_{1} {\vec{𝒖}}_{1} + α_{2} {\vec{𝒖}}_{2} \\ ⋮ \\ \vec{𝒐} & = & α_{0} {\vec{𝒖}}_{0} + α_{1} {\vec{𝒖}}_{1} + α_{2} {\vec{𝒖}}_{2} + \dots + α_{k - 1} {\vec{𝒖}}_{k - 1} \\ \vec{𝒐} & = & α_{0} {\vec{𝒖}}_{0} + α_{1} {\vec{𝒖}}_{1} + α_{2} {\vec{𝒖}}_{2} + \dots + α_{k - 1} {\vec{𝒖}}_{k - 1} + α_{k} {\vec{𝒖}}_{k} \end{matrix}

Set $a_{i} = \frac{α_{i}}{α_{k}}$ to obtain

\begin{matrix} \vec{𝒐} & = & a_{0} {\vec{𝒖}}_{0} + a_{1} {\vec{𝒖}}_{1} + a_{2} {\vec{𝒖}}_{2} + \dots + a_{k - 1} {\vec{𝒖}}_{k - 1} + {\vec{𝒖}}_{k} \\ = & a_{0} L^{0} \vec{𝒖} + a_{1} L^{1} \vec{𝒖} + a_{2} L^{2} \vec{𝒖} + \dots + a_{k - 1} L^{k - 1} \vec{𝒖} + L^{k} \vec{𝒖} \\ = & (a_{0} L^{0} + a_{1} L^{1} + a_{2} L^{2} + \dots + a_{k - 1} L^{k - 1} + L^{k}) \vec{𝒖} \\ = & (L - λ_{k} I_{n}) (L - λ_{k - 1} I_{n}) \dots (L - λ_{2} I_{n}) (L - λ_{1} I_{n}) \vec{𝒖} \end{matrix}

\begin{matrix} \vec{𝒐} \neq & {\vec{𝒛}}_{0} & = & \vec{𝒖} \\ {\vec{𝒛}}_{1} & = & (L - λ_{1} I_{n}) {\vec{𝒛}}_{0} = (L - λ_{1} I_{n}) \vec{𝒖} \\ {\vec{𝒛}}_{2} & = & (L - λ_{2} I_{n}) {\vec{𝒛}}_{1} = (L - λ_{2} I_{n}) (L - λ_{1} I_{n}) \vec{𝒖} \\ ⋮ \\ {\vec{𝒛}}_{i} & = & (L - λ_{i} I_{n}) {\vec{𝒛}}_{i - 1} \\ ⋮ \\ \vec{𝒐} = & {\vec{𝒛}}_{k} & = & (L - λ_{k} I_{n}) {\vec{𝒛}}_{k - 1} \\ = & (L - λ_{k} I_{n}) (L - λ_{k - 1} I_{n}) \dots (L - λ_{2} I_{n}) (L - λ_{1} I_{n}) \vec{𝒖} \end{matrix}

The number $λ$ is an eigenvalue for matrix $L$ if and only if

det (L - λ I_{n}) = 0

The characteristic polynomial of $L$ is

p (λ) = det (L - λ I_{n})

The characteristic equation of $L$ is

det (L - λ I_{n}) = 0

The eigenspace of transformation $𝝋$ associated with eigenvalue $λ$ is

𝑽_{λ} = {\vec{𝒖} \in 𝑽 ∣ 𝝋 (\vec{𝒖}) = λ \vec{𝒖}}

The eigenspace is a subspace.

For $p (x) = {(x - λ_{1})}^{m_{1}} {(x - λ_{2})}^{m_{2}} \dots {(x - λ_{t})}^{m_{t}}$

Eigenvectors with distinct eigenvalues are linearly independent.

Two matrices $A$ and $B$ are similar if there is matrix $S$ such that

A = S^{-1} B S

A matrix is diagonalizable if and only if it is similar to a diagonal matrix. A matrix that is not diagonalizable is called defective.

An $n \times n$ matrix is diagonalizable if and only if it has $n$ linearly independent eigenvectors.