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Mathematics LibreTexts

12: Eigenvalues and Eigenvectors

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    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,

    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).