Page 85 - Applied Numerical Methods Using MATLAB

74 SYSTEM OF LINEAR EQUATIONS
row vectors. The remaining component of the solution x o

T
o
T −1
o
T −1
o
T
x o− = x − x o+ = x − A [AA ] b = x − A [AA ] Ax o
T
T −1
= [I − A [AA ] A]x o
is in the null space N(A), since it satisfies Eq. (2.1.4). Note that
T
T −1
P A = [I − A [AA ] A]
is called the projection operator.
2. The solution (2.1.7) can be obtained by applying the Lagrange multiplier
method (Section 7.2.1) to the constrained optimization problem in which
2
we must find a vector x minimizing the (squared) norm ||x|| subject to the
equality constraint Ax = b.
Eq.(7.2.2) 1 2 T 1 T T
Min l(x, λ) = ||x|| − λ (Ax − b) = x x − λ (Ax − b)
2 2
By using Eq. (7.2.3), we get

∂ T T T T −1
J = x − A λ = 0; x = A λ = A [AA ] b
∂x
∂ T T −1
J = Ax − b = 0; AA λ = b; λ = [AA ] b
∂λ

Example 2.1. Minimum-Norm Solution. Consider the problem of solving the
equation

x 1
[ 12 ] = 3; Ax = b, where A = [ 12 ], b = 3 (E2.1.1)
x 2

This has infinitely many solutions and any x = [ x 1 x 2 ] T satisfying this
equation, or, equivalently,

1 3
x 1 + 2x 2 = 3; x 2 =− x 1 + (E2.1.2)
2 2
is a qualified solution. Equation (E2.1.2) describes the solution space as depicted
in Fig. 2.1.
On the other hand, any vector in the row space of the coefficient matrix A
can be expressed by Eq. (2.1.3) as

1
+ T
x = A α = α (α is a scalar, since M = 1) (E2.1.3)
2