# eigenvectors - Eigenvalues Eigenvectors Example Suppose...

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Eigenvalues & Eigenvectors Example Suppose . Then . So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y -axis. We observe that and . Thus, vectors on the coordinate axes get mapped to vectors on the same coordinate axis. That is,

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for vectors on the coordinate axes we see that and are parallel or, equivalently, for vectors on the coordinate axes there exists a scalar so that . In particular, for vectors on the x -axis and for vectors on the y -axis. Given the geometric properties of we see that has solutions only when is on one of the coordinate axes. Definition Let A be an matrix. We call a scalar an eigenvalue of A provided there exists a nonzero n -vector x so that . In this case, we call the n -vector x an eigenvector of A corresponding to . We note that is true for all in the case that and, hence, is not particularly interesting. We do allow for the possibility that . Eigenvalues are also called proper values (“eigen” is German for the word “own” or “proper”) or characteristic values or latent values . Eigenvalues were initial used by Leonhard Euler in 1743 in connection with the solution to an order linear differential equation with constant coefficients. Geometrically, the equation implies that the n -vectors are parallel.
Example Suppose . Then is an eigenvector for A corresponding to the eigenvalue of as . In fact, by direct computation, any vector of the form is an eigenvector for A corresponding to . We also see that is an eigenvector for A corresponding to the eigenvalue since .

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eigenvectors - Eigenvalues Eigenvectors Example Suppose...

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