Constrained positive deÖnite matrixes
Example
Solvex 1 x 2 !min,x 12 +x 22 = 1.The Lagrangian is
L(x 1 ,x 2 ,λ)=x 1 x 2 +λx 12 +x 22 1.
The necessary conditions
x 2 + 2 λx 1 = 0 ,x 1 + 2 λx 2 = 0.
Ifx 1 =0 thenx 2 =0 which is not a solution. Sox 16 = 0 ,x 26 = 0.
x 2
x 1 = ^2 λ,x 1
x 2 = ^2 λ,x 2
x 1 =x 1
x 2 ,x
12 =x 22
Hence the four solutions are
1
p
2,
1
p
2,
1
p
2,
1
p
2,
1
p
2,
1
p
21
p
2,
1
p
2