14.4: Eigen-Stuff in a Nutshell
- Page ID
- 22931
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A Matrix and its Eigenvector
The reason we are stressing eigenvectors (Section 14.2) and their importance is because the action of a matrix \(A\)on one of its eigenvectors \(\boldsymbol{v}\) is
- extremely easy (and fast) to calculate
\[A \boldsymbol{v}=\lambda \boldsymbol{v} \label{14.5} \]
just multiply \(\boldsymbol{v}\) by \(\lambda\).
- easy to interpret: \(A\) just scales \(\boldsymbol{v}\), keeping its direction constant and only altering the vector's length.
If only every vector were an eigenvector of \(A\)....
Using Eigenvectors' Span
Of course, not every vector can be ... BUT ... For certain matrices (including ones with distinct eigenvalues, \(\lambda\)'s), their eigenvectors span \(\mathbb{C}^n\), meaning that for any \(\boldsymbol{x} \in \mathbb{C}^n\), we can find \(\left\{\alpha_{1}, \alpha_{2}, \alpha_{n}\right\} \in \mathbb{C}\) such that:
\[\boldsymbol{x}=\alpha_{1} v_{1}+\alpha_{2} v_{2}+\ldots+\alpha_{n} v_{n} \label{14.6} \]
Given Equation \ref{14.6}, we can rewrite \(A \boldsymbol{x}=\boldsymbol{b}\). This equation is modeled in our LTI system pictured below:
\[\boldsymbol{x}=\sum_{i} \alpha_{i} v_{i} \nonumber \]
\[b=\sum_{i} \alpha_{i} \lambda_{i} v_{i} \nonumber \]
The LTI system above represents our Equation \ref{14.5}. Below is an illustration of the steps taken to go from \(\boldsymbol{x}\) to \(\boldsymbol{b}\).
\[\boldsymbol{x} \rightarrow\left(\boldsymbol{\alpha}=V^{-1} \boldsymbol{x}\right) \rightarrow\left(\Lambda V^{-1} \boldsymbol{x}\right) \rightarrow\left(V \Lambda V^{-1} \boldsymbol{x}=\boldsymbol{b}\right)\nonumber \]
where the three steps (arrows) in the above illustration represent the following three operations:
- Transform \(\boldsymbol{x}\) using \(V^{-1}\) - yields \(\alpha\)
- Action of \(A\) in new basis - a multiplication by \(\Lambda\)
- Translate back to old basis - inverse transform using a multiplication by \(V\), which gives us \(\boldsymbol{b}\)