Page 189 - DMTH502_LINEAR_ALGEBRA

Unit 14: Invariant Subspaces

2
(a) Prove that the only subspaces of R invariant under T are R and the zero subspace. Notes
2
(b) If U is the linear operator on C , the matrix of which in the standard ordered basis
2
is A, show that U has 1-dimensional invariant subspaces.
2. Let W be an invariant subspace for T. Prove that the minimal polynomial for the restriction
operator T divides the minimal polynomial for T, without referring to matrices.
W
14.3 Summary

 In this unit the idea of invariant subspace of a linear operator T on the n dimension space
helps in introducing a restriction operator Tw as well as a conductor of a vector   V into
the subspace W.

 These concepts generally help us in the diagonalizing of the matrix of the linear
operator T.

 These concepts also lead to triangular form of the matrix A of the linear operator T if A is
diagonalizable.

14.4 Keywords

Invariant: If T is any linear operator on V, then V is invariant under T, as is the zero subspace.
The range of T and the null space of T are also invariant under T.
Restriction Operator: By introducing the concepts of the restriction operator T and the conductor
w
of a vector into the invariant sub-space the characteristic polynomial of the linear operator is
cast into a form where the matrix of T can be seen to be diagonalizable or not.
Restriction: T induces a linear operator on W, called restriction to W.

14.5 Review Questions

1. Show that for the matrix A
 2  2  4
 
A   1 3 4
 
 1  2  3
 
A = A.
2
Find the characteristic values of A.

2. Show that every matrix A such that A = A is similar to a diagonal matrix.
2
14.6 Further Readings

Books Kenneth Hoffman and Ray Kunze, Linear Algebra
Michael Artin, Algebra

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