Math 260, Fall 2018, Assignment 15

I must study politics and war that my sons may have liberty to study mathematics and philosophy.

- John Adams, letter to Abigail Adams, May 12, 1780

Carefully define the following terms, then give one example and one non-example of each:

1. $E_{\lambda}(A)$ (the $\lambda$-eigenspace of the matrix $A$).
2. Characteristic equation (of the matrix $A$).

Carefully describe the following algorithms:

1. Algorithm to find a basis of $E_{\lambda}(A)$, once $\lambda$ is given.
2. Algorithm to find all eigenvalues and eigenvectors of a matrix. (Note that the first step in this "algorithm," which involves solving an equation, may be quite difficult in practice.)

Solve the following problems:

1. Section 7.2, problems 1, 3, 5, 7, 9, 11, 13, and 15. (In problems 1-13, simply find all real eigenvalues.)
2. Section 7.3, problems 1, 3, 5, 7, 9, 11, 13, and 23. (In problems 1-13, simply find bases for all real eigenspaces.)

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