Page 171 - DMTH502_LINEAR_ALGEBRA

Unit 12: Introduction and Characteristic Values of Elementary Canonical Forms

and that for U, find the characteristic values of each operator, and for each such characteristic Notes
value c find a basis for the corresponding space of characteristic vectors.

1 4 1 1
A = , A =
2 3 1 1
2. Let T be the linear operator on R which is represented in the standard ordered basis by the
3
matrix

4 2 2
5 3 2
2 4 1
3
Prove that T is diagonalizable by exhibiting a basis for R , each vector of which is a
characteristic vector of T.

3 0 0
3. Let A = 0 2 5
0 1 2

Is A similar over the field R to a diagonal matrix? Is A similar over the field C to a diagonal
matrix?

12.3 Summary

 When a matrix of a linear operator under a certain ordered basis is in the diagonal form
then some properties of the linear operator can be real at a glance on this matrix.
 In this unit the characteristic values and the corresponding characteristic vectors of a
matrix are found which help us in answering the question whether the given matrix is
diagonalizable over the F or not.

12.4 Keywords

Invertible Matrix: The invertible matrix P formed out of the characteristic vectors of a vector A
shows that A and PAP are similar and also PAP is in the diagonal form.
–1
–1
Null Space: If T is any linear operator and c is any scalar, the set of vectors , such that T = c is
a sub-space of V. It is null space of linear transformation (T– cI).

12.5 Review Questions

1. If T be the linear operator on C which is represented in the ordered basis by the matrix
3
1 i 1
A = i 0 0
1 0 0

Prove that T is diagonalizable by exhibiting a basis for C , each vector of which is a
3
characteristic vector of T.

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