Linear Algebra, Theory And Applications, 2012a

7.10. EXERCISES 191
Show that the algorithm cannot converge for this example. Hint: Try a few iterations
of the algorithm.
( ) ( )
0 −1
0 −2
5. Show the two matrices A ≡
and B ≡
are similar; that is
4 0
2 0
there exists a matrix S such that A = S −1 BS but there is no orthogonal matrix
Q such that Q T BQ = A. Show the QR algorithm does converge for the matrix B
although it fails 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 A having
eigenvector z + iw, show that w ≠ 0.
8. Suppose A = Q T 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) mustbereal.
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 have
magnitude equal to 1.
14. Suppose A is an n × n matrix which is diagonally dominant. Recall this means
∑
|a ij | < |a ii |
show A −1 must exist.
j≠i
15. Give some disks in the complex plane whose union contains all the eigenvalues of the
matrix
⎛
⎝ 1+2i 4 2
⎞
0 i 3 ⎠
5 6 7
16. Show a square matrix is invertible if and only if it has no zero eigenvalues.
17. Using Schur’s theorem, show the trace of an n × n matrix equals the sum of the
eigenvalues and the determinant of an n × n matrix is the product of the eigenvalues.
18. Using Schur’s theorem, show that if A is any complex n × n matrix having eigenvalues
{λ i } listed according to multiplicity, then ∑ i,j |A ij| 2 ≥ ∑ n
i=1 |λ i| 2 . Show that equality
holds if and only if A is normal. This inequality is called Schur’s inequality. [19]