If the eigenspace, W (I presume you mean "corresponding to ") has dimension r, then there exist r independent vectors (a basis for W), , such that . You can construct abasisfor V containing those vectors: where n is the dimension of V. Written in terms of that ordered basis, A is diagonal, with on the diagonal, for the first r rows and columns, with some other n- r by n- r matrix for the last n-r rows and columns. The Characteristic polynomial for that matrix obviously has the form where P(x) is an n- r degree polynomial. It is possible that divides P(x) but certainly it dividesat leastr times.