Linear Algebra, Theory And Applications, 2012a

196 SPECTRAL THEORY
could just differentiate the above formula using the fundamental theorem of calculus
and verify it works. Another way is to assume the solution in the form
and find c (t) to make it all work out.
parameters.
x (t) =Φ(t) c (t)
This is called the method of variation of
42. Show there exists a special Φ such that Φ ′ (t) =AΦ(t) , Φ (0) = I, and suppose
Φ(t) −1 exists for all t. Show using uniqueness that
Φ(−t) =Φ(t) −1
and that for all t, s ∈ R
Φ(t + s) =Φ(t)Φ(s)
Explain why with this special Φ, the solution to (7.25) can be written as
x (t) =Φ(t − t 0 ) x 0 +
∫ t
t 0
Φ(t − s) f (s) ds.
Hint: Let Φ (t) be such that the j th column is x j (t) where
Use uniqueness as required.
x ′ j = Ax j , x j (0) = e j .
43. You can see more on this problem and the next one in the latest version of Horn
and Johnson, [16]. Two n × n matrices A, B are said to be congruent if there is an
invertible P such that
B = PAP ∗
Let A be a Hermitian matrix. Thus it has all real eigenvalues. Let n + be the number
of positive eigenvalues, n − , the number of negative eigenvalues and n 0 the number of
zero eigenvalues. For k a positive integer, let I k denote the k × k identity matrix and
O k the k × k zero matrix. Then the inertia matrix of A is the following block diagonal
n × n matrix.
⎛
⎝ I n +
I n−
O n0
⎞
⎠
O n0
Show that A is congruent to its inertia matrix. Next show that congruence is an
equivalence relation. Finally, show that if two Hermitian matrices have the same
inertia matrix, then they must be congruent. Hint: First recall that there is a
unitary matrix, U such that
⎛
⎞
U ∗ AU = ⎝ D n +
D n−
⎠
where the D n+ is a diagonal matrix having the positive eigenvalues of A, D n− being
defined similarly. Now let ∣ ∣
∣D n− denote the diagonal matrix which replaces each entry
of D n− with its absolute value. Consider the two diagonal matrices
⎛
⎞
Dn −1/2
+
∣∣Dn− ∣ −1/2
Now consider D ∗ U ∗ AUD.
D = D ∗ =
⎜
⎝
I n0
⎟
⎠