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 : edikrnd@gmail.com
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
be taken to be unitary, A and B are said to be
unitarily consimilar.
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
referred to different bases in complex vector
spaces and the theory of consimiliarity of
complex matrix plays an important role in
quantum mechanics.
This paper presents necessary and sufficient
conditions for condiagonalization of a square
complex matrix and gives some examples of
reducing a square complex matrix to diagonal
form by consimilarity.
Let denotes the set of all real numbers, be the
set of all complex numbers, for
is the
mxn
conjugate of x. F
denotes the set of all mxn
matrices on a field F , the conjugate of A. We
write
if A is similar to B,
if A is
consimilar to B, and
permuation similar to B.
If
and
S=U is unitary then
if S=Q is complex
orthogonal then
, if S=R a
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
statement is true as well, and that the diagonal
matrix can always be taken to be nonnegative.
Thus the problem of unitary condiagonalizable
has also been solved already in the following
theorem.
if A is
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
Theorem 2.1([1]). A matrix
is unitarily
condiagonalizable if and only if it is symmetric.
Definition 2.3([3]). Let
there exist
and
be given. If
such that
Then is said to be coneigenvalue of A and x is
said to be coneigenvector of A corresponding to .
Example 1 Let
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.
By (1) the real representation
complex matrix A is
0

0

A 
1

0
of a square
1 1 0

0 0 1
0 0 -1 

1 0 0
And the characteristic polynomial of
.
is
4. Condiagonalization by Real Representation
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 .
We arise to the most important theorem for
reducing of a square complex matrix by real
representation and its algorithm.
Theorem 4.1([2]). Let
. Then A is a
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.
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.
Algorithm
For
, let
denotes the
characteristic polinomial of square complex
matrix. Explanation of condiagonalization is
expressed by the following propositon
Proposition
3.1([2]).
Let
be
condiagonalizable square complex matrix, then
1.
2. The eigenvalues of the real representation
matrix
are real, the nonzero real
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
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
the ordinary sense but is not condiagonalizable.
Example
,
In which
have
. Since
(2)
is diagonalizable in
4
Matrix
is
neither
diagonalizable nor condiagonalizable. How about
we
matrix
, is condiagonalizable ? if yes,
you should find real diagonal matrix J
consimilar to H.
that
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
Algebra. Prentice Hall, New jersey, 1991.