Group Theory and Linear Algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 21 September 2014

Lecture 10: Eigenvectors and annihilators

Let V be a vector space over . Let f:VV be a linear transformation.

The [t]-module defined by f is the vector space V with [t]-action given by p·v=(a0+a1f++aNfN)v if vV and p=a0+a1t+a2t2++aNtN[t].

V= { (a1a2a3a4a5) |ai } has basis B= { (10000), (01000), (00100), (00010), (00001), } and the matrix Bf= ( 21000 02100 00200 00051 00005 ) defines f:VV. Then (3t+6t3)· (a1a2a3a4a5) = ( 3 ( 21 21 2 51 5 ) +6 ( 21 21 2 51 5 ) 3 ) (a1a2a3a4a5) =

An f-invariant subspace, or [t]submodule, of V is a subspace WV such that if wW then fwW.

Let λ. The λ-eigenspace of f is Vλ={vV|fv=λv}. An eigenvector with eigenvalue λ is a vector vVλ.

In our previous example, ( 21 21 2 51 5 ) (70000)= (140000)= 2·(70000). So (70000)V2. ( 21 21 2 51 5 ) (00010)= (00050)= 5· (00010). So (00010)V5. ( 21 21 2 51 5 ) (00100)= (01200) 2· (00100). So (00100) V2.

The λ-generalized eigenspace of V is Vλgen= {vV|there existsk>0with(f-λ)kv=0}. In our previous example, f-2 = ( 21 21 2 51 5 ) -2 ( 1 1 1 1 1 ) = ( 01 01 0 31 3 ) , (f-2)2 = ( 001 00 0 96 9 ) , (f-2)3 = ( 000 00 0 81108 81 ) and (f-2)3 (a1a2a300) = (00000) if a1,a2,a3. So V2gen { (a1a2a300) |a1,a2,a3 } .

The annihilator of V is annV= { p[t]| ifvVthenpv=0 } . The minimal polynomial of f is m[t] such that annV=m[t] where m[t]={mq|q[t]}={multiples ofm}.

In our previous example, (f-2)3 = ( 000 00 0 81108 81 ) , (f-5)2 = ( 9-61 9-6 9 00 00 ) and (f-2)3 (f-5)2= ( 000 000 000 00 00 ) . So (t-2)3(t-5)2annV.

Let f:VV be a linear transformation.

(a) Let λ. Then Vλ is an f-invariant subspace of V.
(b) Let λ. Then Vλgen is an f-invariant subspace of V.
(c) If p1,p2annV then p1+p2annV.
(d) If pannV and q[t] then pqannV.

Proof.

Notes and References

These are a typed copy of Lecture 10 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on August 16, 2011.

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