Math 260, Fall 2019, 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. Eigenvector (for an $n\times n$ matrix $A$).
2. Eigenvalue.
3. $E_\lambda$ (the $\lambda$-eigenspace of $A$).
4. Eigenbasis.

Carefully state the following theorems (you do not need to prove them):

1. Characteristic equation ("If $\lambda$ is an eigenvalue of $A$, then $\mathrm{det}(A-\lambda I)=\dots$")

Carefully describe the following algorithms:

1. Procedure to compute a basis for $E_\lambda$ (note that $E_\lambda$ is the kernel of a certain matrix, so this is not really a new procedure).
2. Procedure to find an eigenbasis, if one exists.

Solve the following problems:

1. Section 7.3, problems 1, 11, 13, 15, 17, and 19 (find an eigenbasis if one exists; you may ignore the instruction concerning "diagonalization").

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