Introduction
Back in 2015 I published an article [1] about the QR decomposition of a centrosymmetric real matrix. In 2016 I started thinking about the meaning of this decomposition and centrosymmetric matrices in general. I discovered that centrosymmetric matrices have a natural interpretation as split-complex matrices and even more, that the centrosymmetric QR decomposition of a matrix actually corresponds to a QR decompositon of a split-complex matrix for which the original matrix is it representation. In a certain sense, the centrosymmetric QR decomposition introduced in [1] of a real square matrix of order
is equivalent to a QR decomposition of a corresponding split-complex square matrix of order
. All these notions will be made precise in the following sections. This blog post is based on my own preprint.
Matrix representations
If
is a finite-dimensional vector space then we denote by
the set of all linear transformations
. Recall that if
is an
-dimensional vector space and
an ordered basis for
, then every linear transformation
has an
matrix representation with respect to
denoted
. Further, for any two linear transformations
we have
. The standard ordered basis for
i.e. the basis
is defined as
if
and
otherwise.
An algebra
is an ordered pair
such that
is a vector space over a field
and
is a bilinear mapping called multiplication.
Let
be an algebra. A representation of
over a vector space
is a map
such that
for all
.
Let
denote the set of
real matrices. If
is an
-dimensional vector space and
an ordered basis for
then every linear transformation
has a matrix representation
. For each
we have
. Since
is
-dimensional, we have and ordered basis
and
. A matrix representation of
with respect to
} is a map
such that
for all
. Further, we have
for all
. These are all well known notions from representation theory, for further information, one can consult one of the standard textbooks, for example see [3].
Algebra of split-complex numbers
Let
, where
is given by
(1) 
for all

. It is straightforward to verify that

is
an algebra. This is the well known algebra of
split-complex numbers. The split-complex numbers are also sometimes known as
hyperbolic numbers.
Similarly as for the complex numbers, each real number

can be identified with the pair

.
With this correspondence, the pair

has the property

and

. Due to this property,

is called the
hyperbolic unit.
Since

, in the following we shall denote a pair

simply with

. The
conjugate of

is defined as

.
For a hyperbolic number
we define the real part as
and hyperbolic part as
.
For the module we set
and we have
for all
.
For an extensive overview of the theory of hyperbolic numbers as well of their usefulness in Physics one can check the literature, for example [2]. For the rest of this blog post, we shall refer to these numbers as split-complex numbers.
Centrosymmetric representation of split-complex matrices
By
we denote the set of all
split-complex matrices i.e. matrices in which entries are split-complex numbers. Note that
if and only if there exist
real matrices
,
such that
. If
is a matrix then its transpose is defined as
for all
and is denoted with
. In the following we denote by
the
identity matrix and by
the
zero matrix. Let
be defined as
for each
. Note that
. A matrix
is upper-triangular if
for all
.
A real matrix
is centrosymmetric if
. An overview of centrosymmetric matrices can be found in [4]. We denote by
the set of all
centrosymmetric real matrices.
For the algebra
of split-complex numbers the well-known matrix representation
with respect to
is given by
(2) 
It is straightforward to check that for all

we have

.
Further, on the vector space
there is a natural multiplication operation
given by
(3) 
for all

. It is easy to verify that

is an algebra. In the following we refer to this algebra as the
algebra of split-complex (square) matrices and denote it with

.
Note that in the following whenever we have two matrices
, their product shall explicitly be written with a dot '
', e.g.
to indicate multiplication defined in (3). Otherwise, if
we simply write
.
To state and prove our main result, we shall need the following well known characterization of centrosymmetric matrices.
Proposition 1:
Let
. Then
if and only if there exist
such that
(4) 
Proof:
Suppose
![Rendered by QuickLaTeX.com \[A =\begin{bmatrix}X & Y \\W & Z\end{bmatrix}.\]](https://konrad.burnik.org/wordpress/wp-content/ql-cache/quicklatex.com-dfc22a1c4d521ab6264a5e940d168b8b_l3.png)
Since

is centrosymmetric, we have

, or equivalently, in block-form
*** QuickLaTeX cannot compile formula:
\[\begin{bmatrix}\ & J \\J & \\end{bmatrix}\begin{bmatrix}X & Y \\W & Z\end{bmatrix} =\begin{bmatrix}X & Y \\W & Z\end{bmatrix}\begin{bmatrix}\ & J \\J & \\end{bmatrix}\]
*** Error message:
\begin{bmatrix} on input line 17 ended by \end{equation*}.
leading text: ...begin{bmatrix}\ & J \\J & \\end{bmatrix}\]
Missing $ inserted.
leading text: ...begin{bmatrix}\ & J \\J & \\end{bmatrix}\]
Missing } inserted.
leading text: ...begin{bmatrix}\ & J \\J & \\end{bmatrix}\]
Missing \cr inserted.
leading text: ...begin{bmatrix}\ & J \\J & \\end{bmatrix}\]
Missing $ inserted.
leading text: ...begin{bmatrix}\ & J \\J & \\end{bmatrix}\]
Missing } inserted.
leading text: ...begin{bmatrix}\ & J \\J & \\end{bmatrix}\]
Missing \right. inserted.
leading text: ...begin{bmatrix}\ & J \\J & \\end{bmatrix}\]
Improper \prevdepth.
leading text: \end{document}
Missing $ inserted.
leading text: \end{document}
Missing } inserted.
leading text: \end{document}
This is equivalent to
![Rendered by QuickLaTeX.com \[\begin{bmatrix}JW & JZ \\JX & JY\\end{bmatrix} =\begin{bmatrix}YJ & XJ \\ZJ & WJ\end{bmatrix}\]](https://konrad.burnik.org/wordpress/wp-content/ql-cache/quicklatex.com-dd5627f494d4bca7581085523a98716c_l3.png)
We now have

and

, so
![Rendered by QuickLaTeX.com \[A = \begin{bmatrix}X & JWJ \\W & JXJ\end{bmatrix}\]](https://konrad.burnik.org/wordpress/wp-content/ql-cache/quicklatex.com-7c3957443d30be0edce1f4d5805ced92_l3.png)
Now, by choosing

and

and from the fact

it follows that

has the form (4).
Conversely, suppose
has the form (4). It can easily be shown by block-matrix multiplication that
, hence
is centrosymmetric.
QED.
The map
defined as
(5) 
is a matrix representation of

. We call the representation

the
standard centrosymmetric matrix representation of

.
Proof:
Let
and
be such that
and
.
We now have

which proves the claim.
QED.
Proposition.
Let
. Then
.
Proof.
Let
. Then
![Rendered by QuickLaTeX.com \[{cs}(Q^T) = {cs}(A^T + jB^T) = \begin{bmatrix}A^T & B^T J \\J B^T & J A^T J\end{bmatrix}.\]](https://konrad.burnik.org/wordpress/wp-content/ql-cache/quicklatex.com-50052d6274db4c924f8f8c3e0721d654_l3.png)
On the other hand, keeping in mind that

we have
![Rendered by QuickLaTeX.com \[{cs}(Q)^T = \begin{bmatrix}A & B J \\J B & JAJ\end{bmatrix}^T =\begin{bmatrix}A^T & (JB)^T \\(BJ)^T & (JAJ)^T\end{bmatrix}= \begin{bmatrix}A^T & B^T J \\J B^T & J A^T J\end{bmatrix}.\]](https://konrad.burnik.org/wordpress/wp-content/ql-cache/quicklatex.com-9714f4b56d03557826eb2b0e6f887b47_l3.png)
Hence,

.
QED.
Proposition.
The map
is a bijection.
Proof.
Injectivity. Let
and
and
. From this,
it follows that
or
. Assume that
. Then

Since

we have

. Let now

and assume that

. Then from

it follows

. Now multiplying

with

from the left implies

, which is a contradiction.
We conclude that

is injective.
Surjectivity. Let
. By proposition 1 we can find matrices
and
such that (4) holds. But then
and since
was arbitrary, we conclude that
is surjective.
Now, injectivity and surjectivity of
imply by definition that
is a bijection.
QED.
Correspondence of QR decompositions
Definition:
Let
. A pair
with
is a QR decomposition of
over
if the following holds:
is orthogonal, i.e.
,
is upper-triangular,

The notion of a
double-cone matrix was introduced by the author in [1].
Here we state the definition in block-form for the case of
.
Definition:
Let
. Then
is a double-cone matrix iff there exist
both upper-triangular such that
![Rendered by QuickLaTeX.com \[H = \begin{bmatrix}A & BJ \\JB & JAJ\end{bmatrix}.\]](https://konrad.burnik.org/wordpress/wp-content/ql-cache/quicklatex.com-808405c728b3915d63f5080e9576518b_l3.png)
Definition:
Let
. A pair
, with
is a centrosymmetric QR decomposition of
if the following holds:
is orthogonal matrix,
is double-cone matrix,

The algorithm to obtain an approximation of a centrosymmetric QR decomposition of a given centrosymmetric matrix
was given in [1].
The following theorem provides one interpretation of the centrosymmetric QR decomposition, in the case of square centrosymmetric matrices of even order by establishing the equivalence of their centrosymmetric QR decomposition with the QR decomposition of the corresponding split-complex matrix.
Theorem 1 (QR decomposition correspondence):
Let
. Then
is a QR decomposition of
if and only if
![Rendered by QuickLaTeX.com \[(\CsEmb{Q}, \CsEmb{R}) \in \CsMat{2n} \times \CsMat{2n}\]](https://konrad.burnik.org/wordpress/wp-content/ql-cache/quicklatex.com-e9a458e3f996c705150a04da9762d369_l3.png)
is a centrosymmetric QR decomposition of

.
Proof.
Let
be a QR decomposition of
.
Let
and
. We have

Since

it follows that

.
From this we have

i.e.

hence

is orthogonal. Since

is upper-triangular and

, then by definition we have that both

and

are upper-triangular. Further,

is centrosymmetric by definition. From this it follows that

is centrosymmetric double-cone.
Finally, we have

. Hence,

is
a centrosymmetric QR decomposition of

.
Conversely, let
. If
is a centrosymmetric QR decomposition of
then
where
is centrosymmetric and orthogonal and
is a double-cone matrix.
From the fact that
is centrosymmetric we have (by Proposition 1) that
![Rendered by QuickLaTeX.com \[W = \begin{bmatrix}W_1 & W_2 J \\J W_2 & J W_1 J\end{bmatrix}\]](https://konrad.burnik.org/wordpress/wp-content/ql-cache/quicklatex.com-c10ff9bad6cc022b85b4befe7f99a024_l3.png)
Now the property of

being orthogonal i.e. the condition

implies
(6) 
On the other hand, we have
(7) 
First we prove that

is orthogonal. From (6) we obtain

The matrix

is centrosymmetric and double-cone which implies
![Rendered by QuickLaTeX.com \[Y = \begin{bmatrix}Y_1 & Y_2 J \\J Y_2 & J Y_1 J\end{bmatrix}\]](https://konrad.burnik.org/wordpress/wp-content/ql-cache/quicklatex.com-5788a9be2737c3666186cb5481c37365_l3.png)
where both

and

are upper-triangular. This now implies that

is upper-triangular.
Finally, let us prove that
. We have
![Rendered by QuickLaTeX.com \[Q \cdot R = {cs}^{-1} (W) {cs}^{-1} (Y) = {cs}^{-1}(WY) = {cs}^{-1}(\CsEmb{A}) = A.\]](https://konrad.burnik.org/wordpress/wp-content/ql-cache/quicklatex.com-dd022056b2f5c502b64e158d2a87d15e_l3.png)
We conclude that

is a QR decomposition of

.
QED.
Example:
Let
![Rendered by QuickLaTeX.com \[A = \begin{bmatrix}1 + 2j & 2 + 3j \\3 + 4j & 4 + 5j\end{bmatrix}.\]](https://konrad.burnik.org/wordpress/wp-content/ql-cache/quicklatex.com-db6205a4ba7d1af37703a787ba0df9c9_l3.png)
Note that

where
(8) 
We have
![Rendered by QuickLaTeX.com \[{cs}(A) = {cs}(W + jZ) = \begin{bmatrix}W & ZJ \\JZ & JWJ\end{bmatrix}= \begin{bmatrix}1 & 2 & 3 & 2 \\3 & 4 & 5 & 4 \\4 & 5 & 4 & 3 \\2 & 3 & 2 & 1\end{bmatrix}\]](https://konrad.burnik.org/wordpress/wp-content/ql-cache/quicklatex.com-1c69ab9d656b5f8fe677e72f6445a893_l3.png)
By applying the

algorithm from [1] to

we obtain the approximations:

Applying
to
and
yields:

with

. Now, from Theorem 1 we conclude that

is an approximation of a QR decomposition of

.
Conclusion
We introduced the standard centrosymmetric representation
for split-complex matrices. Using this representation we proved that a QR decomposition of a square split-complex matrix
can be obtained by calculating the centrosymmetric QR decomposition introduced by the author in [1] of its centrosymmetric matrix representation
.
References
- Burnik, Konrad. A structure-preserving QR factorization for centrosymmetric real matrices,
Linear Algebra and its Applications 484(2015) 356 - 378
- Catoni, Francesco and Boccaletti, Dino and Cannata, Roberto and Catoni, Vincenzo and Zampeti, Paolo, Hyperbolic Numbers. Geometry of Minkowski Space-Time Springer Berlin Heidelberg Berlin, Heidelberg, (2011) 3–23 ISBN: 978-3-642-17977-8
- Curtis, Charles W.; Reiner, Irving. Representation Theory of Finite Groups and Associative Algebras, John Wiley & Sons (Reedition2006 by AMS Bookstore), (1962) ISBN 978-0-470-18975
- James R. Weaver. Centrosymmetric (Cross-Symmetric) Matrices,Their Basic Properties, Eigenvalues, and Eigenvectors, The American Mathematical Monthly, Mathematical Association of America, (1985) 10-92, 711–717 ISSN: 00029890, 19300972 doi:10.2307/2323222
QR_decomposition_of_split_complex_matric
Copyright © 2018, Konrad Burnik