1. Find the spectra and eigenvetors for the two matrices below. Show your work.

A =

?

?

3 5 3

0 4 6

0 0 1

?

? A =

?

?

a 1 0

1 a 1

0 1 a

?

?

2. Find the matrix A for each of the indicated linear transformation y = Ax. Find its eigenvalues

and eigenvectors.

(a) Reflection about the x-axis in R2

. Here x = [x y].

(b) Orthogonal projection of R3 onto the plane x = y. Here x = [x y z].

3. Prove that trace of a square real or complex matrix A equals the sum of its eigenvalues. This

fact is often a useful check on the accuracy of eigenvalue calculations. Demonstrate with an

example of your choosing.

4. Prove that the eigenvectors of a real symmetric matrix corresponding to different eigenvalues

are orthogonal.

5. Do there exist real symmetric 3 × 3 matrices that are orthogonal (except for the unit matrix

I)?

6. Prove that Hermitian, skew-Hermitian and unitary matrices are all normal matrices.

7. Find the similarity transformation that diagonalizes the following matrix. Show details of your

work.

A =

?

?

16 0 0

48 -8 0

84 -24 4

?

?

8. Use the power method to find the largest eigenvalue to 5 significant figures of the first matrix

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