6: Eigenvalues and Eigenvectors
- Page ID
- 138354
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We have often explored new ideas in matrix algebra by making connections to our previous algebraic experience. Adding two numbers, \(x + y\), led us to adding vectors \(\vec{x}+\vec{y}\) and adding matrices \(A + B\). We explored multiplication, which then led us to solving the matrix equation \(A\vec{x}=\vec{b}\), which was reminiscent of solving the algebra equation \(ax = b\).
This chapter is motivated by another analogy. Consider: when we multiply an unknown number \(x\) by another number such as \(5\), what do we know about the result? Unless, \(x = 0\), we know that in some sense \(5x\) will be “\(5\) times bigger than \(x\).” Applying this to vectors, we would readily agree that \(5\vec{x}\) gives a vector that is “\(5\) times bigger than \(\vec{x}\).” Each entry in \(\vec{x}\) is multiplied by \(5\).
Within the matrix algebra context, though, we have two types of multiplication: scalar and matrix multiplication. What happens to \(\vec{x}\) when we multiply it by a matrix \(A\)? Our first response is likely along the lines of “You just get another vector. There is no definable relationship.” We might wonder if there is ever the case where a matrix – vector multiplication is very similar to a scalar – vector multiplication. That is, do we ever have the case where \(A\vec{x}=a\vec{x}\), where \(a\) is some scalar? That is the motivating question of this chapter.
- 6.1: Eigenvalues and Eigenvectors
- This page explores eigenvalues and eigenvectors, defining and illustrating their significance through matrix multiplication and applications across various fields. It outlines the process of finding these values using the characteristic polynomial, demonstrating with examples of 2x2 and 3x3 matrices. Specific eigenvalues and their corresponding eigenvectors are provided for multiple matrices.
- 6.2: Properties of Eigenvalues and Eigenvectors
- This page discusses eigenvalues and eigenvectors of matrices, highlighting their relationships, particularly between a matrix, its transpose, and its inverse. It explains that \(A\) and \(A^T\) share eigenvalues, and presents the Eigenvalue Theorem related to matrix invertibility. Examples illustrate eigenvalue calculations, including cases with repeated and complex eigenvalues, and underscore the significance of zero eigenvalues.
Thumbnail: A 2×2 real and symmetric matrix represent a stretching and shearing of the plane. The eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on them. (CC0; Jacopo Bertolotti via Wikipedia)