Linear Algebra/Eigenvalues and Eigenvectors

In this subsection we will focus on the property of Corollary 2.4.

Template:Anchor("Eigen" is German for "characteristic of" or "peculiar to"; some authors call these characteristic values and vectors. No authors call them "peculiar".)

That example shows why the "non-$\overrightarrow{0}$" appears in the definition. Disallowing $\overrightarrow{0}$ as an eigenvector eliminates trivial eigenvalues.

The next example illustrates the basic tool for finding eigenvectors and eigenvalues.

Problem 11 checks that the characteristic polynomial of a transformation is well-defined, that is, any choice of basis yields the same polynomial.

Notice the familiar form of the sets of eigenvectors in the above examples.

By the lemma, if two eigenvectors ${\overrightarrow{v}}_1$ and ${\overrightarrow{v}}_2$ are associated with the same eigenvalue then any linear combination of those two is also an eigenvector associated with that same eigenvalue. But, if two eigenvectors ${\overrightarrow{v}}_1$ and ${\overrightarrow{v}}_2$ are associated with different eigenvalues then the sum ${\overrightarrow{v}}_1+{\overrightarrow{v}}_2$ need not be related to the eigenvalue of either one. In fact, just the opposite. If the eigenvalues are different then the eigenvectors are not linearly related.

Exercises

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References

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