Page 190 - DMTH502_LINEAR_ALGEBRA

Linear Algebra Richa Nandra, Lovely Professional University

Notes Unit 15: Simultaneous Triangulation and
Simultaneous Diagonalization

CONTENTS
Objectives

Introduction
15.1 Simultaneous Triangulation and Simultaneous Diagonalization
15.2 Summary
15.3 Keywords

15.4 Review Question
15.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Know the structure of the triangular form of a matrix of a linear operator T on a space V
over the field F.
 Understand that we can diagonalize two or more commuting matrices simultaneously.
 Know that the matrix of a linear operator T commutes with that of a polynomial of a linear
operator T.

Introduction

In this unit we are again exploring the properties of a linear operator on the spaceV over the field F.
In an upper triangular or lower triangular matrix the elements in the diagonal are the
characteristic values.

15.1 Simultaneous Triangulation and Simultaneous Diagonalization

Let V be a finite-dimensional space and let  be a family of linear operators on V. We ask when
we can simultaneously triangulate or diagonalize the operators in , i.e., find one basis  such
that all of the matrices [T], T in , are triangular (or diagonal). In the case of diagonalization,
it is necessary that  be a commuting family of operators: UT = TU for all T, U in . That follows
from the fact that all diagonal matrices commute. Of course, it is also necessary that each operator
in  be a diagonalizable operator. In order to simultaneously triangulate, each operator in 
must be triangulable. It is not necessary that  be a commuting family; however that condition
is sufficient for simultaneous triangulation (if each T can be individually triangulated). These
results follow from minor variations of the proofs of Theorems 1 and 2 of unit 14.
The subspace W is invariant under (the family of operators)  if W is invariant under each
operator in .
Lemma: Let  be a commuting family of triangulable linear operator on V. Let W be a proper
subspace of V which is invariant under . There exists a vector  in V such that

(a)  is not in W;
(b) for each T in , the vector T is in the subspace spanned by  and W.

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