eigenvectors - Eigenvalues & Eigenvectors Example...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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,
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 2
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 .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 17

eigenvectors - Eigenvalues & Eigenvectors Example...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online