Recent content by buzzmath

I know that the square roots of the nonzero eigenvalues of P*P or PP* are the singular values of P and thus also P*. The ||P|| = max{singular value} thus P and P* have the same 2-norm. ||P||=||P*||=1. I write <(PP*-P*P)x,x> = <P*x,P*x>-<Px,Px>=||P*x||^2-||Px||^2 ||P|| = sup (||Px||) where ||x||...

Homework Statement Let P be a projection. The definition used is P is a projection if P = PP. Show that ||P|| >=1 with equality if and only if P is orthogonal. Let ||.|| be the 2-normHomework Equations P = PP. P is orthogonal if and only if P =P*The Attempt at a Solution I've proved the...

Is that true? What about A = [1,0;0,0]? This is not zero or I but A^2 = A

I'm still confused because we're not using the fact that ||P|| = 1 just the fact that <(P-P*P)x,Px> = <(P-P*P)x,P*Px> and that <(P-P*P)x,Px> =||(P-P*P)x|| - <(P-P*P)x,P*Px> so ||(P-P*P)x|| = 0 and thus P-P*P = 0 thus P = P* but we never use ||P|| = 1 so wouldn't this work for and finite...

so I have <(P-P*P)x,Px> = <(P-P*P)x,(P-P*P)x>+<(P-P*P)x,P*PX> I know <Px,Px> = 1 = x*P*Px now I write <(P-P*P)x,Px>=x*P*Px-x*PP*Px=1-x*PP*Px and <(P-P*P)x,(P-P*P)x> = ||(P-P*P)x|| and <(P-P*P)x,P*PX> = x*P*P*Px-x*PP*P*Px = 1-x*PP*Px Then <(P-P*P)x,Px> = 1-x*PP*Px =...

Yes, I see the problem now. I was thinking I could use ||.|| = 0 only is . = 0 but <Px-P*Px,Px> = 0 doesn't mean that the first part is zero. I've been playing with this and can't find how to use the fact that ||P|| = 1 any pointers? Thanks

So I understand now that (P-P*)P=0 implies P=P* since P is nonzero. I'm questioning my method for getting there now. The reason is I don't use the fact that ||P|| = 1 I only use the fact that <Px,Px>=<P*Px,Px> For example What if ||P|| = 2 then <Px,Px> = <Px,PPx> = <P*Px,Px> = 2 and <Px,Px> -...

Homework Statement P is mxm complex matrix, nonzero, and a projector (P^2=P). Show 2-norm ||P|| >= 1 with equality if and only if P is an orthogonal projector (P=P*) Homework Equations Let ||.|| be the 2-norm The Attempt at a Solution a. show ||P|| >= 1 let v be in the range...

Is this a typo? if so how would I go about solving the problem using the method I've started? or any other way? Thanks

Homework Statement An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims. If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed? [b]2. Homework Equations [/]...

this is just for undergrad too. If you majored in math and wanted to go to grad school in physics I don't think it would be that hard. I'm a math guy and the upper level classes can hard but a lot is just hard work and experience. Which ever one you choose you're almost certain to take upper...

Laplace equation is when f(x,y)=0. f(x,y) can represent many things physically. the solution of this problem can represent many things for example u could be a steady state temperature of the cross section of a rod with an electrical current.

I know that the series isn't zero but they say that the series goes to infinity which I don't understand since the sequence goes to zero. I mean how would they come up with this? is there something I'm not seeing? What do you mean by doing a change of coordinates? how would I get integral from...

Homework Statement 1. f is a function defined on the interval -a<x<a and has Fourier coefficients an=0 bn=1/n^(1/2) what can you say about the integral from -a to a of f^2(x)dx? 2. Show that as n goes to infinity the Fourier sine coefficients of the function f(x)=1/x -pi<x<pi tend to a...

Homework Statement Is the derivative of a periodic function periodic? Is the integral of a periocic function periodic? Homework Equations a function if periodic if f(x)=f(x+p) The Attempt at a Solution I think that the derivative won't be periodic but the integral will. I'm not...