8.66 (a), §2 Solution

Since A^T=A, the matrix is symmetric.

Eigenvalues:

There are two roots: and

The eigenvector corresponding to satisfies

Taking v_1y = -2, then v_1x = 1, giving an eigenvector (1,-2). Normalizing this vector to length one gives:

The eigenvector corresponding to satisfies

Taking v_2y = 1, then v_2x = 2, giving after normalization:

Finally:

Check:

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