- #1
TheFerruccio
- 220
- 0
Homework Statement
Find the matrix norm.
Homework Equations
[itex]\left(
\begin{array}{cc}
4 & 2 \\
2 & 1 \\
\end{array}
\right)[/itex]
The Attempt at a Solution
There are definitely different ways to solve this. I will use Lagrange multipliers.
[itex]\textbf{A}x =
\left(
\begin{array}{cc}
4x_1+2x_2 \\
2x_1+x_2 \\
\end{array}
\right)[/itex]
[itex](\left|\textbf{A}\textbf{x}\right|)^2=(4x_1+2x_2)(4x_1+2x_2)+(2x_1+x_2)(2x_1+x_2)=18x_1^2+20x_1 x_2+5x_2^2[/itex]
Use Lagrange multipliers at this step, with the condition that the norm of the vector we are using is x. The goal is to find the unit vector such that A maximizes its scaling factor.
[itex]f = 18x^2+20x_1x_2+5x_2^2+\mu(x_1^2+x_2^2)[/itex]
Find the derivatives in the ::x_1:: and ::x_2:: directions and set each to 0.
[itex]\frac{\partial f}{\partial x_1}=36x_1+20x_2+2\mu x_1=0[/itex]
[itex]\frac{\partial f}{\partial x_2}=20x_1+10x_2+2\mu x_2=0[/itex]
I am not sure where to go from here. I know that the norm of the matrix is 5, and I tried comparing coefficients to get different values for ::\mu:: and pick the right ::\mu:: for the right unit vector. If I were to directly compare coefficients, I get two different values for ::\mu::.
First value:
[itex]\mu = -18[/itex]
Second value:
[itex]\mu = -5[/itex]
These are associated with two unit vectors which would make the the two partial derivative equations hold true:
(1,0)
and
(0,1)
These are definitely wrong, because if I use the first unit vector (1,0), the associated magnitude of Ax would be root 18. If I were to use the second unit vector (0,1), the associated magnitude would be root 5.
What am I doing wrong?