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,

$string.jpg$

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

David Cherney, Tom Denton, and Andrew Waldron (UC Davis)