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# 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).