How Do Rank Properties of Idempotent Matrices Affect Their Sum?

[mpegwmvavi]

Homework Statement

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

Homework Equations

no

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?

 

[Dick]

You want to show that you can choose a basis of eigenvectors. Then rank(A) is the number of eigenvectors with eigenvalue 1 and rank(A-I) is the number of eigenvectors with eigenvalue 0, right?

 

[mpegwmvavi]

From equation (a), since we have rankA=r, the dimension of solution space,or in another words, the dimension of null space = nullityA = n-r by fundamental theorem of linear systems. From the same way, we have rank(A-I)=s, so nullity(A-I)=n-s.

Now, if the equation hold,ie,if :rankA+rank(A-I) = nullityA + nullity(A-I),I could complete my proof,ie,r+s=n
but the problem is that I don't get why the equation above,r+s=(n-s)+(n-r),holds.
Could somebody help me,please. I've stuck in this for a long long long~~~~~~~time. Thx.

 

[209511979]

how can i express (-9-7x-15x^2) as a linear combination of F1 (2+x+4x^2) , F2 (1-x+3x^2) and F3 (3+2x+5x^2).

 

[Mark44]

how can i express (-9-7x-15x^2) as a linear combination of F1 (2+x+4x^2) , F2 (1-x+3x^2) and F3 (3+2x+5x^2).

I realize you are new to Physics Forum, but when you tack a totally unrelated problem onto the end of an existing thread, it is called "highjacking" the thread. Instead of adding onto an existing thread, you should start a new thread.

Also, when you post here, you need to provide the complete problem statement (what you have is OK for that), any relevant equations or formulas, and your attempt at the solution.

What does a "linear combination" of functions mean?

 

[209511979]

how can i find an equation for the plane spanned by the vectors V1=(-1,1,1) and V2=(3,4,4).

 

[Dick]

From equation (a), since we have rankA=r, the dimension of solution space,or in another words, the dimension of null space = nullityA = n-r by fundamental theorem of linear systems. From the same way, we have rank(A-I)=s, so nullity(A-I)=n-s.

Now, if the equation hold,ie,if :rankA+rank(A-I) = nullityA + nullity(A-I),I could complete my proof,ie,r+s=n
but the problem is that I don't get why the equation above,r+s=(n-s)+(n-r),holds.
Could somebody help me,please. I've stuck in this for a long long long~~~~~~~time. Thx.

You really didn't pay enough attention to post 2.