Hello Again, Linear Algebra: A Second Look at the Subject through a Collection of Exercises and Solutions by Pavel Grinfeld

Author:Pavel Grinfeld [Grinfeld, Pavel]
Language: eng
Format: azw3
Publisher: Insightful Texts
Published: 2014-01-12T05:00:00+00:00

Among the pairs of vectors in the figure, can you identify those that satisfy the eigenvalue relationship ? In other words, those pairs in which one vector is a scalar multiple of the other? Yes you can: it's the vectors that lie along (and appear on top of each other as a single vector) and those perpendicular to ! Call those vectors and , and and . These pairs of vectors satisfy

Thus, we have found the eigenvectors and and the corresponding eigenvalues and . And we did it by pure geometric reasoning without writing down a single number, let alone a matrix. The main point, however, is not so much the mechanics of what we did, but the very idea is that the concept of eigenvalues and eigenvectors is geometric!

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