Total potential energy in matrix

[omgitsroy326]

Consider the matrix A=[3 -1; -1 3]. (a) Can the quadratic form x'*A*x evaluate to zero for some non-zero vector x? (x is a 2x1 column vector, and x' means x transposed: usual Matlab notation). (b) Does the equation A*x=b, b arbitrary, have a unique solution? If it does, prove it. (c) Show that the total potential energy f(x)=1/2*x'*A*x-b'*x has a minimum for x that solves the linear equations A*x=b.

This is the question which I'm approached with
my answer is as following
a) No, because it's a symetric matrix only x'Ax = 0 only if vector x = 0.
b) Yes , it is not underdetermined and det does not equal 0
c) iono

i was wondering what you guys thought

 

[AKG]

a) Compute $x^tAx$ and you will obtain an expression in terms of the entries of x, and this expression, as is given, will be a quadratic form. With this expression, you should easily be able to see that unless x is the zero vector, the expression will not be equal to zero (if x is to have real entries).

b) Compute det(A). Note that it is non-zero. Therefore, A is invertible, so there exists an inverse of A, $A^{-1}$. If $Ax = b$, then $A^{-1}Ax = A^{-1}b$, so $x = A^{-1}b$, so clearly x exists and is unique.

c) The best way I can think to do this is to compute f(x) to obtain an expression in terms of the entries of x. Essentially, f is a function from $\mathbb{R}^2$ to $\mathbb{R}$. Compute the Jacobian of f at x, and find the x such that the Jacobian is zero. For all those points, compute the Hessian, and show that there is only one that has a positive definite Hessian. Calculate the value of f at this point. Also, calculate the value of $f(A^{-1}b)$, and show that the two values are the same, thereby showing that the vector where f has it's minimum is indeed the unique solution to $Ax = b$.

If you are unfamiliar with any of the terminology above, you can easily look it up at wikipedia.com or mathworld.com, and then, if you're still stuck, ask for further clarification.

 

[omgitsroy326]

wow.. thanks.. that was very detailed