Given the Generalized Rayleigh Quotient
Q(x) = (Ax dot x)/(Bx dot x)
and that A and B are both real n x n matrices, and B is positive definite.
How would I go about proving that Q(x) has at least one critical point. In class we restricted ||x|| to 1 so that ||x||^2 also is 1, but we never actually see ||x||^2 in this equation, so it doesn't seem like that would be helpful. If someone can please help me with where to start that'd be awesome! I know that taking the gradient is the first step, but after that I'm utterly confused.
Thank you for that explanation. Also, how do you make the nice equations show up instead of what I had shown above?
And is matrix Py a column vector? Because to do the dot product we need two column vectors. And why can we go from (Py)^tB(Py) to y^Py? Why can we change the order of y^t and P^t and why does the B change to A?
The next thing I need to do is prove that x is a CP of Q if and only if Ax = (lambda)Bx for some real lambda. I see that you've already proved one portion of this, but I don't know how I'd go about proving the other part?
Thanks!
. The same for .
Yes, it is, is a column vector, so is also a column vector.And is matrix Py a column vector? Because to do the dot product we need two column vectors.
We can't. It isAnd why can we go from (Py)^tB(Py) to y^Py?
Well known property: .Why can we change the order of y^t and P^t
A typo on my part. I've just corrected it.and why does the B change to A?
Thanks! Those clarifications are great help!
Last thing, could you help me prove that if all the above given information is true, and that at a critical point x it's true that , that the determinant of . I don't know how I'd go about showing this. Not sure if I said this, but x can't be 0.