Automatic Control Systems Homework Proof on Linear Algebra

[bcilek]

Homework Statement

Show that the geometric multiplicity (denoted by Qi) associated with an eigenvalue(lamda i)

of an nxn matrix A can also be expressed as Qi=n-rank(A-(Idendity Matrix)*(lamda i)).

P.S.:: Lamda i is simply a scaler quantity since it is an eigenvalue where A and I(Idendity Matrix) are matrices.

I could not prove this.I can simply try an example and verify that this is correct but our instructor wants us a matematical proof..Thanks very much for now your valuable help...

 

[fzero]

Do you know how the multiplicity is related to the dimension of the space of solutions to

$$(A - \lambda_i I_n)u = 0?$$ (*)

You might want to choose the basis of eigenvectors $$\{ \mathbf{v}_k \}$$ to compute this. The space of solutions to (*) is called the kernel of the matrix $$A - \lambda_i I_n$$. The space of vectors that don't satisfy (*) are called the image of $$A - \lambda_i I_n$$. The dimension of the image of a matrix is equal to the rank of the matrix. There is a simple relationship between the dimension of the vector space and the dimensions of the kernel and image of a matrix that you can use.