MHB Cutiee pie's question at Yahoo Answers (dimension of eigenspaces).

[Fernando Revilla]

Here is the question:

Give an easy example of a 3x3 matrix with characteristic equation (1-lambda)^3=0 such that:?
a) the eigenspace corresponding to lambda=1 has dimension one
b) the eigenspace corresponding to lambda=1 has dimension two
c) the eigenspace corresponding to lambda=1 has dimension three

Here is a link to the question:

Give an easy example of a 3x3 matrix with characteristic equation (1-lambda)^3=0 such that:? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello cutiee pie,

Choose:

(i) $A=\begin{bmatrix}{1}&{1}&{0}\\{0}&{1}&{1}\\{0}&{0}&{1}\end{bmatrix}$ (ii) $A=\begin{bmatrix}{1}&{0}&{1}\\{0}&{1}&{0}\\{0}&{0}&{1}\end{bmatrix}$ (iii) $A=\begin{bmatrix}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{1}\end{bmatrix}$

Clearly, in these cases the characteristic polynomial is $\chi(\lambda)=(1-\lambda)^3$. Besides,

(i) $\dim V_1=3-\mbox{rank }(A-I)=3-\mbox{rank }\begin{bmatrix}{0}&{1}&{0}\\{0}&{0}&{1}\\{0}&{0}&{0}\end{bmatrix}=3-2=1$(ii) $\dim V_1=3-\mbox{rank }(A-I)=3-\mbox{rank }\begin{bmatrix}{0}&{0}&{1}\\{0}&{0}&{0}\\{0}&{0}&{0}\end{bmatrix}=3-1=2$

(iii) $\dim V_1=3-\mbox{rank }(A-I)=3-\mbox{rank }\begin{bmatrix}{0}&{0}&{0}\\{0}&{0}&{0}\\{0}&{0}&{0}\end{bmatrix}=3-0=3$