Let P be a n times n matrix satisfying P2 P Show that KerP

Let P be a n times n matrix satisfying P^2 = P. Show that Ker(P) intersect Range(P) is {0}. Show that 0 and 1 are the only possible eigenvalues for P. Identify the eigenspaces E_0 and E_1.

Solution

4a. Let x be in ker(P) and range(P)

Hence,

P(x)=0

And x is in range P

so for some y P(y)=x

P(P(y))=P(y)=x=P(x)=0

HEnce, x=0

4.b

Let, Px=kx

P^2x=Px=kPx=k^2x

And so on

So, k,k^2,k^3,.. are all eigenvalues

But P is only for size nxn

SO, k=1 or 0

c.

Let k=0

P(x)=0

Hence, E_0=ker(P)

k=1

P(x)=x

P^2(x)=P(x)=x

Hence,

E_1=set of fixed points of the map.