ON THE TRANSFORMATION AND REAL REPRESENTATION OF A SQUARE
COMPLEX MATRIX TO DIAGONAL FORM BY CONSIMILARITY
Edi Kurniadi
Department of Mathematics, Mathematics and Natural Sains Faculty, Padjadjaran University, Bandung-Jatinangor Indonesia, 45363
Email : [email protected]
Abstract
A square complex matrix A is said to be condiagonalizable if there exist a nonsingular S such that diagonal. This paper, by means of real representation of a square complex matrix, studies algebraic technique of reducing a square complex matrix to diagonal form by consimilarity. Besides, this paper not only tells information about when a square complex matrix can be reduced to diagonal form by unitary consimilarity transformation but also gives algorithm for condiagonalization a square complex matrix via real representation.
Key words: Consimilarity, condiagonalization, coneigenvector.
AMS(2000) subject clasifications 15A21
1. Introduction
The study of condiagonalization of a square complex matrix was motivated by the existing theory of diagonalization and similarity of square real matrix. Some notions and results could be formulated by transfering the corresponding knowledge from similarity to consimilarity and from diagonalization to condiagonalization. Consimilarity of square complex matrix arises as a result of studying antilinear transformation
permuation similar to
B
.
2. Consimilarity and Condiagonalization
Definition 2.1([1]). Two matrices are
said to be consimilar if there exist a nonsingular
such that . If the matrix S can real nonsingular matrix then
Definition 2.2([1]). A matrix is said to be condiagonalizable if S can be chosen so that is diagonal. It is said to be unitarily condiagonalizable if it can be reduced by consimilarity to the required form via a unitary matrix.
If is unitarily condiagonalizable then
for some unitary
and . Thus,
, and
hence A is symmetric. The converse of the above
has also been solved already in the following theorem.
Theorem 2.1([1]). A matrix is unitarily condiagonalizable if and only if it is symmetric.
Definition 2.3([3]). Let be given. If
there exist and such that
Then is said to be coneigenvalue of A and x is
said to be coneigenvector of A corresponding to .
Theorem 2.2([1]). Let be given. If has k distinct nonnegative eigenvalues then A has at least k independent coneigenvectors. If k = n, A is condiagonalizable. If k = 0, A has no coneigenvectors at all.
The remaining problem concerning condiagonalization is to characterize usefully those matrices that can be condiagonalized by a consimilarity that is not necessarily unitary. In this paper we study characterizations of condiagonalization of a square complex matrix by means of real representation, derive an algorithm of reducing a square complex matrix to diagonal form by consimilarity.
3. Real Representation of a Square Complex Matrix
Let , A can uniquely written by as
A=A1+A2i, . Define real
representation matrix
(1)
The real representation matrix is called real
representation of A. Here is a
coneigenvalue of A if and only if are
eigenvalues of .
For , let denotes the
characteristic polinomial of square complex matrix. Explanation of condiagonalization is expressed by the following propositon
eigenvalues of appear in positive pairs and
the 0 eigenvalue of appears in pairs.
Proof If A is a condiaginalizable matrix then
there exists a nonsingular complex matrix S such
that condiagonalizable matrix if and only if is a diagonalizable matrix and
Theorem 4.1 tells us about necessary and sufficient conditions of condiagonalization of a square complex matrix and gives an algorithm for condiagonalizations.
Algorithm for condiagonalization[2] Let
be a square complex matrix
Step 1 Find the real representation of a square
complex matrix A.
Step 2 Find the characteristic polynomial of real
representation and its all real
eigenvalues .
Step 3 Construct real diagonal matrix J.
Example 2 From example 1 above the
eigenvalues of real representation are 1, 1, -1,
-1. Since diagonalizable matrix, so by theorem
4.1 A is condiagonalizable matrix and by theorem
4.1 . Note that A in example
1 above is condiagonalizable but not diagonalizable in the ordinary sense.
Example 3 Matrix is diagonalizable in
the ordinary sense but is not condiagonalizable.
Example 4 Matrix is neither
diagonalizable nor condiagonalizable. How about
matrix , is condiagonalizable ? if yes,
you should find real diagonal matrix J that
consimilar to H.
5. CONCLUSIONS
In this study, an algorithm for condiagonalization is developed to reduce a square complex matrix by consimilarity. Theorem 2.1 and 2.2 give us information when a given square complex matrix
A can be reduced to diagonal form by
transformation for nonsingular S. For
further researching about consimilarity and condiagonalization of a family of matrices, you should find some applications in quantum mechanics.
REFERENCES
[1] Horn R A, Johnson C R. Matrix Analysis.
Cambridge University Press, New York, 1985.
[2] Jiang tongsong, Wei Musheng. On the Reduction of a Complex Matrix to Triangular or Diagonal by Consimilarity. A Journal of Chinese Universities. Numerical Math., 2006, vol.15, pp 107-112.
[3] Qingchun Li, Shugong Zhang. The Inclusion Interval of Basic Coneigenvalues of a Matrix. Preprint, 2005.
[4] Bernard K, David R H. Elementary Linear