Page 211 - DMTH502_LINEAR_ALGEBRA

Unit 18: The Primary Decomposition Theorem

2. If N is a nilpotent operator on an n-dimensional vector space V, show that the characteristic Notes
n
polynomial for N is x .
18.3 Summary

 The primary decomposition theorem is based on the fact that the minimal polynomial of
the linear operator is the product of the irreducible.
 This helps in finding the projection operates which are polynomials in T.

 The direct decomposition of the vector space V in terms of the invariant subspaces helps in
inducing linear operators T on these subspaces W .
i i
 The induced operator T on W by T has the minimal polynomial as well as due to the
i i
factorisation of the minimal polynomial of T.

18.4 Keywords

Invariant Sub-spaces: If a vector  in V is such that  and T are in the subspace W of V then W
is invariant subspace of V over the field F.

Nilpotent Transformation: A nilpotent transformation N on the vector space V represented by
K
a matrix A is such that A = 0 for some integer K and A  0. Here K is the index of nilpotency.
K–1
Projection Operators: The projection operator E acting on the vector  gives E =  for the
i i i i
2
subspace W and gives zero for other. Also E  E and E E = 0 for i  j
i i i i j
18.5 Review Questions

3
1. Let T be the linear operator on R which is represented by the matrix
 3 1  1
 
A  2 2  1
 
 2 2 0 
 
in the standard ordered basis. Show that T = D + N where D is a diagonalizable operator
and N a nilpotent vector.
3
2. Show that the linear operator T on R represented by the matrix
 1  1 1 
 
A  2  2 2
 
 1 1  1
 
is nilpotent.

18.6 Further Readings

Books Kenneth Hoffman and Ray Kunze, Linear Algebra
I.N. Herstein, Topics in Algebra

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