# Characteristic Polynomial Question

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• October 16th 2010, 02:18 AM
BlackadderVII
Characteristic Polynomial Question
Hi,
Really unsure about how to do this question, so any help/answers/links to a similar thread would be appreciated.

landa is an eigenvalue of a linear map T:V to V (V over C). Suppose the eigenspace W has dimension r. Prove that the characteristic polynomial CA(x) of a nxn matrix A representing T is divisiblr by ((landa)-x)^r, and deduce that r is less than or equal to the multiplicity of landa as an eigenvalue of A.

Thankyou
• October 16th 2010, 08:50 AM
HallsofIvy
If the eigenspace, W (I presume you mean "corresponding to $\lambda$") has dimension r, then there exist r independent vectors (a basis for W), $\{v_i\}$, such that $Av_i= \lambda v_i$. You can construct a basis for V containing those vectors: $\{v_1, v_2, ..., v_r, u_1, ..., u_{n-r}\}$ where n is the dimension of V. Written in terms of that ordered basis, A is diagonal, with $\lambda$ 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 $(x- \lambda)^r P(x)$ where P(x) is an n- r degree polynomial. It is possible that $x- \lambda$ divides P(x) but certainly it divides $(x- \lambda)^rP(x)$ at least r times.
• October 16th 2010, 09:08 AM
BlackadderVII
Thankyou HallsofIvy, you've been most usefull.