Principles of Mathematics in Operations Research

4.5 Problems 65

Properties:

i. A 6 CmXn, 3 unitary M — U 9 U~lAU = T is upper-triangular. The

eigen values of A must be shared by the similar matrix T and appear

along the main diagonal.

ii. Any Hermitian matrix A can be diagonalized by a suitable U.

Definition 4.5.5 The matrix N is called normal if NNH = NHN. Only for

normal matrices, T = U"^1 NU = A where A is diagonal.

Problems

4.1. Determinant

Prove property 11 in Section 4.1.2.

4.2. Jordan form

Let A =

1 1-1-1 -1

2 112 1

0 110-1

1-113 1

2-2224

Find S such that S^AS =

"2 1

2

2 1

2

2_

Hint:

Choose V2 € M[{A - XI)^2 }, v\ = [A - A/]w 2. Similarly, choose v± and

Finally, choose v* e M"UA - XI)}.

v 3 -

Finally, choose v 5 6 N[{A - XI)}.

4.3. Using Jordan Decomposition

Let A =

• 1
  10
  0
  0

I

10

l

10

0

0 '

l

10

I

10.

Find A^10.

4.4. Differential Equation System
Let the Blue (allied) forces be in a combat situation with the Red (enemy)
forces. There are two Blue units (Xi, X 2 ) and two Red military units (Yi, Y^)-
At the start of the combat, the first Blue unit has 100 (X° = 100) combatants,
the second Blue unit has 60 (X° = 60) combatants. The initial conditions
for the Red force are Y° = 40 and Y 2 ° = 30. Since the start of the battle
(t = 0), the number of surviving combatants (less than the initial values due
to attrition) decrease monotonically and the values are denoted by X\, X\,
Y{, and Y.

The first Blue unit is subjected to directed fire from all the Red forces,
with an attrition rate coefficient of 0.03 Blue 1 targets/Red 1 firer per unit
time and 0.02 Blue 1 targets/Red 2 firer per unit time. The second Blue unit
is also subjected to directed fire from all the Red forces, with an attrition rate