Introductory Linear Algebra/Eigenvalues and eigenvectors

Motivation

Before discussing Template:Colored em and Template:Colored em, we provide some motivations to them. Template:Colored example Template:Colored example We can see from this example that for some special matrices, their powers can be computed conveniently, by expressing in the form of $PD{P}^{-1}$ in which $P$ is invertible matrix and $D$ is diagonal matrix.

Naturally, given a matrix, we would be interested in knowing whether it can be expressed in the form of $PD{P}^{-1}$, and if it can, what are $P$ and $D$, so that we can compute its power conveniently. This is the main objective of this chapter.

Eigenvalues, eigenvectors and diagonalization

In view of the motivation section, we have the following definition. Template:Colored definition Template:Colored remark Template:Colored example Template:Colored exercise The following are important and general concepts, which is related to diagonalizability in some sense. Template:Colored definition Template:Colored remark Template:Colored example Template:Colored exercise The following theorem relates diagonalizable matrix with eigenvectors and eigenvalues. Template:Colored theorem

Proof. In the following, we use $\left(\begin{array}{ccc}{𝐯}_1& \cdots & {𝐯}_n\end{array}\right)$ to denote the matrix with columns ${𝐯}_1,\dots ,{𝐯}_n$, in this order. $$\begin{array}{cccc}& & A& =PD{P}^{-1}\\ & \iff & AP& =PD\underset{I}{\underbrace{P{P}^{-1}}}\\ & \iff & A\left(\begin{array}{ccc}{𝐯}_1& \cdots & {𝐯}_n\end{array}\right)& =\left(\begin{array}{ccc}{𝐯}_1& \cdots & {𝐯}_n\end{array}\right)\left(\begin{array}{cccc}{\lambda }_1& 0& \cdots & 0\\ 0& {\lambda }_2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\lambda }_n\end{array}\right)\\ & \iff & \left(\begin{array}{ccc}A{𝐯}_1& \cdots & A{𝐯}_n\end{array}\right)& =\left(\begin{array}{ccc}{\lambda }_1{𝐯}_1& \cdots & {\lambda }_n{𝐯}_n\end{array}\right)\\ & \iff & A{𝐯}_1& ={\lambda }_1{𝐯}_1,\dots ,A{𝐯}_n={\lambda }_n{𝐯}_n.\end{array}$$ We have now proved that ${𝐯}_1,\dots ,{𝐯}_n$ are Template:Colored em. It remains to prove that they are Template:Colored em, which is true since they are Template:Colored em if and only if $P$ is invertible by the proposition about relationship between invertibility and linear independence. $◻$

Template:Colored remark Then, we will introduce a convenient way to find Template:Colored em. Before this, we introduce a terminology which is related to this way of finding eigenvalues. Template:Colored definition Template:Colored remark Template:Colored example Template:Colored proposition

Proof. $$\begin{array}{cccc}& & \lambda \text{ is an }& \text{eigenvalue of }A\\ & \iff & A𝐯& =\lambda 𝐯\text{for some }𝐯\ne \mathrm{𝟎}\\ & \iff & (A-\lambda I_n)𝐯& =\mathrm{𝟎}\text{for some }𝐯\ne \mathrm{𝟎}\\ & \iff & A-\lambda I_n& \text{ is non-invertible}\text{by simplified invertible matrix theorem}\\ & \iff & \det (A-\lambda I_n)& =0\end{array}$$

$◻$

Then, we will introduce a concept which is related to Template:Colored em. Template:Colored exercise Template:Colored definition Template:Colored remark After introducing these terminologies and concepts, we have the following algorithmic procedure for a diagonalization of an $n\times n$ matrix:

1. compute all Template:Colored em of $A$ by solving $\det (A-\lambda I)=0$
2. for each eigenvalue ${\lambda }_1,\dots ,{\lambda }_k$ of $A$, find a Template:Colored em ${\beta }_1,\dots ,{\beta }_k$ for the Template:Colored em $E_{{\lambda }_1},\dots ,E_{{\lambda }_k}$ respectively
3. if ${\beta }_1,\dots ,{\beta }_k$ together contain $n$ vectors ${𝐯}_1,\dots ,{𝐯}_n$ (if not, $A$ is Template:Colored em diagonalizable), define $$P=\left(\begin{array}{ccc}{𝐯}_1& \cdots & {𝐯}_n\end{array}\right)$$
4. we have $A=PD{P}^{-1}$ in which $D$ is a Template:Colored em matrix whose Template:Colored em entries are Template:Colored em corresponding to ${𝐯}_1,\dots ,{𝐯}_n$

Template:Colored remark Template:Colored example Template:Colored example Template:Colored example Template:Colored example Template:Colored exercise In the following, we will discuss some mathematical applications of diagonalization, namely deducing the sequence formula, and solving system of Template:Colored em (ODEs). Template:Colored example Template:Colored exercise Template:Colored example Template:Colored exercise