SOLUTION: Let P be a projection, so {{{P^2 = P}}}. If {{{c<>1}}}, compute {{{(I-cP)^(-1)}}}.

Question 1062030: Let P be a projection, so . If , compute .
Answer by ikleyn(52814) (Show Source): You can put this solution on YOUR website!
.
Let P be a projection, so . If , compute .
~~~~~~~~~~~~~~~~~~~

I can do it for < 1. Again, let us assume that |c| < 1.

Then   =  = (since ) =  = 


      ( apply the formula for the sum of an infinite geometric progression ) 


      =  =  = 


      =  =  = .

Answer. = .

You may check it by making direct calculations that * = I under the condition = .

And after checking it directly (as I said) you may to extend the formula for |c| > 1. It is valid for |c| > 1, too.

RELATED QUESTIONS
There were fatal errors in the solution of one tutor to this problem, so I'm reposting: (answered by ikleyn)

There were fatal errors in the solution of one tutor to this problem, so I'm reposting: (answered by ikleyn)

Let p(x) be a cubic polynomial. If p(0) = 0, p(1) = 1, p(2) = 2, and p(3) = 3, then... (answered by CPhill,ikleyn,math_tutor2020)

let a,b and c be three mutually and exhaustive events find p(b) if 1/3 p(c)=1/2... (answered by Fombitz)

Multiply {{{ (c+p)(c^2-cp+p^2)... (answered by mananth)

Given that p(A)= 1/4 p(B)= 1/5 p(C)= 1/2 Compute the following 1)... (answered by MathLover1)

Given P(A)=0.2; P(B) = 0.5, P(A|B) = 0.3: (a) Compute P(A and B). (b) Compute P(A or... (answered by jim_thompson5910,spunky)

Let A and B be events with P(A)=1/2, P(B)=1/3 and P(A∩B)=1/4. Find i) P(A | B) ii) (answered by ikleyn,greenestamps)

Let X be b(2, p) and let y be b(4, p). If p(X ≥ 1) = 5/9 , find P(Y ≥... (answered by ikleyn)