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

by vish_maths
Tags: eigen, multiplicity, nullt
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vish_maths
#1
Feb16-13, 09:25 AM
P: 53
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
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micromass
#2
Feb16-13, 12:22 PM
Mentor
micromass's Avatar
P: 18,044
Quote Quote by vish_maths View Post
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 [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.

(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).
vish_maths
#3
Feb16-13, 01:14 PM
P: 53
Thanks a lot micromass. I will do the needful


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