Page 193 - DMTH502_LINEAR_ALGEBRA

Unit 15: Simultaneous Triangulation and Simultaneous Diagonalization

2. Let  be a commuting family of 3 × 3 complex matrices. How many linearly independent Notes
matrices can  contain? What about the n × n case?

15.2 Summary

 In this unit we are dealing with matrices that commute with each other.
 In a triangular matrix the main diagonal has the entries of the characteristic values and it
has not zero entries in the upper part of the diagonal only or non-zero entries in the lower
of the main diagonal.
 If two or more matrices commute then we can diagonalize them simultaneously.

15.3 Keywords

Diagonalizable: Each operator in  is diagonalizable, because its minimal polynomial divides
i
the minimal polynomial for the corresponding operator in .
Ordered Basis: There exists an ordered basis for V such that every operator in  is represented
by a triangular matrix in that basis.

15.4 Review Question

1. Let T be a linear operator on a n-dimension space and suppose that T has n distinct
characteristic values. Prove that any linear operator which commutes with T is a polynomial
in T.

Answers: Self Assessment

1 2   1  1
1. (a) P    , (b) P   
 0 1   1 1 
2. 3, n

15.5 Further Readings

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

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