Multiplicity of an eigen value , k = dim[ Null(T - k I)^( dim V) ]

vish_maths

I have been reading Linear Algebra done right by Sheldon Axler
I got two conceptual queries :

(1) It states that for a Matrix of an operator T = [ 5 1 ; 0 5 ] ( ; indicates next row )
that dim Null [ (T - 5 I )² ] = multiplicity of the eigen value 5 = 2

However, T - 5 I= [ 0 1 ; 0 0 ]

and (T - 5I )² = [ 0 0 ; 0 0 ]

dim Null [ (T - 5 I )² ] ≠ 2

I am a bit confused about the given result in the book hence. Could anyone please clarify.


(2) Multiplicity of an Eigen value , k = dim[ Null(T - k I)^{dim V} ]


I have been trying to prove this without induction . Any direction please ?

thanks

micromass

But the nullspace of [itex]\left(\begin{array}{cc} 0 & 0\\ 0 & 0\end{array}\right)[/itex] is [itex]\mathbb{R}^2[/itex] (all vectors are sent to 0). And the dimension of this is 2.

Please post this in the homework forum, together with an attempt and relevant results and equations. (whether it really is homework is irrelevant).

vish_maths

Thanks a lot micromass. I will do the needful