Principles of Mathematics in Operations Research

4.5 The Complex case 63

and if we generalize to the matrices

eM = I + At +

{At)^2 (At)^3

2! 3!

If we take the derivative of both sides, we have

+ •

deM T A A^2 {2t) A^3 {it^2 )

= I + A + „. + +—)^—L + -

dt

= A

2! 3!

' , (At)^2 (At)^3

I + At+ ±-J-2! + ^J- + ••• 3!

= Ae At

UA = SAS-^1 ,

At r OAO-I SA^2 S-H^2 SA^3 S~H^3

eAt = / + SAS x + -. + -. +

2!

' , {At)^2 {At)^3

i+At+^-^- + ^-T'—+•

3!

S"^1 = SeAtS~^1.

2! 3!

Thus, we have the following theorem.

Theorem 4.4.8 // A can be diagonalized as( A = SAS-1), then ^ — Au

has the solution u(i) = eAtuo = SeAtS~^1 uo, or equivalently u(i) = aieAl'vi +

• • • + aneXntvn, where a — 5 _1«o-

4.5 The Complex case

In this section, we will investigate Hermitian and unitary matrices. The com-
plex field C is denned over complex numbers (of the form x+iy where x,y E.R
and i^2 — —1) with the following operations:

{a + ib) + {c + id) = {{a + c) + i(b+d)) (a + ib)(c + id) = {{ac-bd) + i(cb+ad)).

Definition 4.5.1 The complex conjugate ofa + ibsC is a + ib = a — ib. See
Figure 4-2.

Properties:

i. (a + ib){c + id) — {a + ib)(c + id),
ii. {a + ib) + (c + id) — {a + ib) + (c + id),
Hi. (a + ib)a + ib = a^2 + b^2 — r^2 where r is called modulus of a + ib.
We have a = y/a^2 + b^2 cos 9 and b = \/a^2 + b^2 sin 9 and

a + ib = \]a? + b^2 {cos9 + ism 9) = re^10 (Polar Coordinates),

where re1,6 — cos 9 + i sin 9.