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

by vish_maths
Tags: eigen, multiplicity, nullt
 P: 47 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
Mentor
P: 16,562
 Quote by 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.
But the nullspace of $\left(\begin{array}{cc} 0 & 0\\ 0 & 0\end{array}\right)$ is $\mathbb{R}^2$ (all vectors are sent to 0). And the dimension of this is 2.

 (2) Multiplicity of an Eigen value , k = dim[ Null(T - k I)dim V ]
Please post this in the homework forum, together with an attempt and relevant results and equations. (whether it really is homework is irrelevant).
 P: 47 Thanks a lot micromass. I will do the needful

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