Hey guys,
Iv been given the function:
V(x,y) = kx^2 + k[(y-x)^2] + ky^2
They say :
Write down a symetric matrix A such that V = (X^T)AX where X = (x,y)^T
T: Transpose of matrix.
Whaaaat the helll???
Thanks.
The want you to find a $\displaystyle 2\times2$ symmetric matrix such that:
$\displaystyle \left[\begin{array}{cc}x&y\end{array}\right]\left[\begin{array}{cc}a&b\\c&d\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]=kx^2+k(y-x)^2+ky^2=2kx^2-2kxy+2ky^2$
Multiplying out that matrix product gives you $\displaystyle ax^2+bxy+cxy+dy^2$. So what do $\displaystyle a,b,c,d$ need to be in order to make the equation hold? Remember that $\displaystyle A$ has to be symmetric.
Thanks man, i did something like that but it didnt come out like that. You just need to take the co-effients of the equation and match it with the equation determined by the multiplication....by the way, how did you know they were looking for a 2x2?
a = 2k
b+c = -2k => b=c=-k
d = 2k
Because of the rules governing matrix multiplication. We were multiplying
$\displaystyle [1\times2]\cdot[m\times n]\cdot[2\times1]$
To multiply the first two, $\displaystyle m$ must be $\displaystyle 2$. To multiply the second pair, $\displaystyle n$ must be $\displaystyle 2$.