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12: Eigenvalues and Eigenvectors

  • Page ID
    1734
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Given only a vector space and no other structure, save for the zero vector, no vector is more important than any other. Once one also has a linear transformation the situation changes dramatically. Consider a vibrating string,

    string.jpg

    whose displacement at point \(x\) is given by a function \(y(x,t)\). The space of all displacement functions for the string can be modelled by a vector space \(V\). At this point, only the zero vector---the function \(y(x,t)=0\) drawn in grey---is the only special vector.

    The wave equation

    \[\frac{\partial^{2} y}{\partial t^{2}}=\frac{\partial^{2} y}{\partial x^{2}}\, ,\]

    is a good model for the string's behavior in time and space. Hence we now have a linear transformation

    \[\left(\frac{\partial^{2} }{\partial t^{2}}-\frac{\partial^{2} }{\partial x^{2}}\right):V\rightarrow V\, .\]

    For example, the function

    \[y(x,t)=\sin t \sin x\]

    is a very special vector in \(V\), which obeys \(L y = 0\). It is an example of an eigenvector of \(L\).

    Thumbnail: Mona Lisa with shear, eigenvector, and grid. Imaged used with permission (Public domain; TreyGreer62).

    Contributor


    This page titled 12: Eigenvalues and Eigenvectors is shared under a not declared license and was authored, remixed, and/or curated by David Cherney, Tom Denton, & Andrew Waldron.

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