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...
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...
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 [/]...
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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?
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Homework Statement
Is the derivative of a periodic function periodic?
Is the integral of a periocic function periodic?
Homework Equations
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The Attempt at a Solution
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