Dears,
I need the solution of the following exercise:
Let V be a real vector space and E an idempotent linear operator on V, i.e.,
a projection. Prove that (I + E) is invertible. Find (I + E)^{-1}. (where I is the identity operator)
Best Regards
Dears,
I need the solution of the following exercise:
Let V be a real vector space and E an idempotent linear operator on V, i.e.,
a projection. Prove that (I + E) is invertible. Find (I + E)^{-1}. (where I is the identity operator)
Best Regards
hint:
suppose A is a matrix such that (I + E)A = I.
then A + EA = I.
applying E to both sides we have:
E(A + EA) = E
EA + E^{2}A = E
but E^{2} = E (since E is a projection) thus:
2EA = E, or EA = (1/2)E.
now substitute this back in the equation
A + EA = I, and solve for A.