Register to reply

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

by vish_maths
Tags: eigen, multiplicity, nullt
Share this thread:
vish_maths
#1
Feb16-13, 09:25 AM
P: 55
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
Phys.Org News Partner Science news on Phys.org
Final pieces to the circadian clock puzzle found
A spray-on light show on four wheels: Darkside Scientific
How an ancient vertebrate uses familiar tools to build a strange-looking head
micromass
#2
Feb16-13, 12:22 PM
Mentor
micromass's Avatar
P: 18,346
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: 55
Thanks a lot micromass. I will do the needful


Register to reply

Related Discussions
What is eigen value, eigen vector etc and what is their physical significance? Linear & Abstract Algebra 8
Eigen Value, Eigen function -Schrodinger wave equation Quantum Physics 1
How i find eigen function and eigen value from equation or either Advanced Physics Homework 3
Need help with deriving the eigen-value and eigen-func for a S-L PDE (1dim Heat EQ) Differential Equations 1
Eigen function, eigen value, eigen vector Quantum Physics 2