MT Exercise-7-정태일

Exercise 7) Find the maximum and the minimum values of the function

$R(\rm\bold x)=\frac{\rm\bold x^{\it T}\it A\rm\bold x}{\rm\bold x^{\it T}\rm\bold x}, \rm\bold x\neq 0$

which is called the Rayleigh quotient of $A$, when

1)

$\begin{bmatrix} 2 &1 &0 &0 \\ 1 &2 &0 &0 \\ 0 & 0 & 4 &0 \\ 0 &0 &0 &1 \end{bmatrix}$

2)

$\begin{bmatrix} 2 &-3 &0 \\ -3 &0 &0 \\ 0 & 0 & 1 \end{bmatrix}$

Sol)

1) Note that the maximum value of $R(\rm\bold x)$ is the maximum eigenvlue of $A$, and similarly for the minimum value.

Reference:http://en.wikipedia.org/wiki/Rayleigh_quotient 2013.12.03

2)