Page 204 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes 17.3 Summary

 In this unit the finite dimensional vector space is decomposed into a direct sum of the
invariant subspaces.

 The linear operator induces a linear operator T on each invariant subspace W by restriction.
i i
 The projection operators can be obtained from the Lagrange polynomials once we know
the characteristic values.

17.4 Keywords

2
Projection Operator: The projection operator E has the property that E = E so its characteristic
values can be equal to 0 and unit.
Restriction: When the finite space V is decomposed into the direct sum of the invariant subspaces
the linear operator induces a linear operator by the process known as restriction.
The Lagrange Polynomials: Help us to find the projection operators for any linear operator T in
terms of the matrix representing T and its characteristic values.
17.5 Review Questions

1. Let T be a linear operator on V. Suppose V = W  ...  W , where each W is invariant
1 k i
under T. Let T be the induced (restriction) operator on W . Prove that the characteristic
i i
polynomial for f is the product of the characteristic polynomials f , f ,..., f .
1 2 k
2. Let T be a linear operator on three dimensional space which is represented by the matrix
 4 2  2
 
A   5 3 2 ,
 
 2 4 1  

Find the matrices E , E , E such that A = C E + C E + C E
1 2 3 1 1 2 2 3 3
E + E + E = I, E E = 0 for i  j
1 2 3 i j
Answers: Self Assessment

1. E = 2I – A, E = A – I, Here E + E = I, A = E + 2E and E E = 0
1 2 1 2 1 2 1 2
2. Here c = 0, c = –2, c = 2
1 2 3
2
E = I – A /4
1
1
I
2
E = (A  2 )A
8
A
3
E = (A  2 )
I
8
17.6 Further Readings

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

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