6.10. EXERCISES 183

could just differentiate the above formula using the fundamental theorem of calculusand verify it works. Another way is to assume the solution in the form

x (t) = Φ (t) c (t)

and find c (t) to make it all work out. This is called the method of variation ofparameters.

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 ∈ RΦ (t+ s) = Φ (t) Φ (s)

Explain why with this special Φ, the solution to 6.26 can be written as

x (t) = Φ (t− t0)x0 +

∫ t

t0

Φ (t− s) f (s) ds.

Hint: Let Φ (t) be such that the jth column is xj (t) where

x′j = Axj , xj (0) = ej .

Use uniqueness as required.

43. You can see more on this problem and the next one in the latest version of Hornand Johnson, [17]. Two n × n matrices A,B are said to be congruent if there is aninvertible P such that

B = PAP ∗

Let A be a Hermitian matrix. Thus it has all real eigenvalues. Let n+ be the numberof positive eigenvalues, n−, the number of negative eigenvalues and n0 the number ofzero eigenvalues. For k a positive integer, let Ik denote the k × k identity matrix andOk the k×k zero matrix. Then the inertia matrix of A is the following block diagonaln× n matrix.  In+

In−

On0

Show that A is congruent to its inertia matrix. Next show that congruence is an equiv-alence relation on the set of Hermitian matrices. Finally, show that if two Hermitianmatrices have the same inertia matrix, then they must be congruent. Hint: Firstrecall that there is a unitary matrix, U such that

U∗AU =

 Dn+

Dn−

On0

where the Dn+ is a diagonal matrix having the positive eigenvalues of A, Dn− beingdefined similarly. Now let

∣∣Dn−

∣∣ denote the diagonal matrix which replaces each entryof Dn− with its absolute value. Consider the two diagonal matrices

D = D∗ =

 D−1/2n+ ∣∣Dn−

∣∣−1/2

In0

