Page 328 - A Course in Linear Algebra with Applications

312 Chapter Nine: Advanced Topics

Exercises 9.1

1. Find unitary or orthogonal matrices which diagonalize the
following matrices:

( a ; !
(i a) (»>(; j -»)

2. Suppose that A is a complex matrix with real eigenvalues
which can be diagonalized by a unitary matrix. Prove that A
must be hermitian.
3. Show that an upper triangular matrix is normal if and only
if it is diagonal.
4. Let A be a normal matrix. Show that A is hermitian if and
only if all its eigenvalues are real.

5. A complex matrix A is called skew-hermitian if A* = —A.
Prove the following statements:
(a) a skew-hermitian matrix is normal;
(b) the eigenvalues of a skew-hermitian matrix are purely
imaginary, that is, of the form a\f^l where a is real;
(c) a normal matrix is skew-hermitian if all its eigenvalues
are purely imaginary.
6. Let A be a normal matrix. Prove that A is unitary if and
only if all its eigenvalues c satisfy \c\ = 1.
7. Let X be any unit vector in C n and put A = I n — 2XX*.
Prove that A is both hermitian and unitary. Deduce that
A = A~\
8. Give an example of a normal matrix which is not hermitian,
skew-hermitian or unitary. [Hint: use Exercises 4, 5, and 6].