(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose A and I are n*n matrices and I is a unit matrix ,and A is an idempotent matrix,ie, A=A^2 .

Show that if rankA=r and rank(A-I)=s,then r+s=n

2. Relevant equations

no

3. The attempt at a solution

I know that if A is an idempotent matrix ,it will have eigenvalues either 0 or 1.

(Proof: Ax=(A^2)x ,and Ax=λx so(A^2)x = A(Ax)=Aλx=λ(Ax)=(λ^2) x

thus, λx=(λ^2) x →(λ-1)λx=0. Suppose x is a nonzero eigenvector, λ = 1 or 0. )

that is, if x1 and x2 are eigenvectors associated with eigenvalue 0 and 1 respectively,

then, A(x1)=0(x1)=0, (a)

and A(x2)=1(x2)=x2

so(A-I)(x2)=0 (b)

Now, I have (a) and (b) , how to show that rankA+rank(A-I)=n?

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# Homework Help: Linear algebra-idempotent

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