340 CHAPTER 13. MATRICES AND THE INNER PRODUCT

which has eigenvalues 3 and 2. I suggest you use a computer algebra system to dothe computations.

4. ↑ Now try the QR algorithm on (0 −12 0

)

Show that the algorithm cannot converge for this example. Hint: Try a few iterationsof the algorithm. Use a computer algebra system if you like.

5. ↑Show the two matrices A≡

(0 −14 0

)and B≡

(0 −22 0

)are similar; that is

there exists a matrix S such that A = S−1BS but there is no orthogonal matrix Q suchthat QT BQ = A. Show the QR algorithm does converge for the matrix B although itfails to do so for A.

6. Let F be an m×n matrix. Show that F∗F has all real eigenvalues and furthermore,they are all nonnegative.

7. If A is a real n× n matrix and λ is a complex eigenvalue λ = a+ ib,b ̸= 0, of Ahaving eigenvector z+ iw, show that w ̸= 0.

8. Suppose A = QT DQ where Q is an orthogonal matrix and all the matrices are real.Also D is a diagonal matrix. Show that A must be symmetric.

9. Suppose A is an n×n matrix and there exists a unitary matrix U such that

A =U∗DU

where D is a diagonal matrix. Explain why A must be normal.

10. If A is Hermitian, show that det(A) must be real.

11. Show that every unitary matrix preserves distance. That is, if U is unitary,

|Ux|= |x| .

12. Show that if a matrix does preserve distances, then it must be unitary.

13. ↑Show that a complex normal matrix A is unitary if and only if its eigenvalues havemagnitude equal to 1.

14. Suppose A is an n×n matrix which is diagonally dominant. Recall this means

∑j ̸=i

∣∣ai j∣∣< |aii|

show A−1 must exist.