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{{Short description|Binary operation}}
In [[mathematics]], the '''Redheffer star product''' is a [[binary operation]] on [[Linear map|linear operators]] that arises in connection to solving coupled [[System of linear equations|systems of linear equations]]. It was introduced by [[Raymond Redheffer]] in 1959,<ref name="Redheffer59" /> and has subsequently been widely adopted in computational methods for [[S-matrix|scattering matrices]]. Given two scattering matrices from different linear scatterers, the Redheffer star product yields the combined scattering matrix produced when some or all of the output [[Scattering channel|channels]] of one scatterer are connected to inputs of another scatterer.

== Definition ==
Suppose <math>A, B</math> are the [[Block matrix|block matrices]]
<math>A = 
\begin{pmatrix}
    A_{11} & A_{12}
    \\
    A_{21} & A_{22}
\end{pmatrix}</math>
and
<math>B =
\begin{pmatrix}
    B_{11} & B_{12}
    \\
    B_{21} & B_{22}
\end{pmatrix} </math>,
whose blocks <math>A_{ij}, B_{kl}</math> have the same shape when
<math>ij = kl</math>.
The Redheffer star product is then defined by:
<ref name="Redheffer59" />

<math>A \star B =

\begin{pmatrix}
    B_{11} (I - A_{12} B_{21})^{-1} A_{11} & B_{12} + B_{11} (I - A_{12} B_{21})^{-1} A_{12} B_{22}
    \\
    A_{21} + A_{22} (I - B_{21} A_{12})^{-1} B_{21} A_{11} & A_{22} (I - B_{21} A_{12})^{-1} B_{22}
\end{pmatrix}
</math>
,

assuming that <math>(I - A_{12} B_{21}), (I - B_{21} A_{12})</math> are [[Invertible matrix|invertible]],
where <math>I</math> is an identity matrix [[Conformable matrix|conformable]]
to <math>A_{12} B_{21}</math> or <math>B_{21} A_{12}</math>, respectively.
This can be rewritten several ways making use of the so-called
[[Woodbury matrix identity|push-through identity]]
<math>(I - A B) A = A (I - B A) \iff A (I - B A)^{-1} = (I - A B)^{-1} A</math>.

Redheffer's definition extends beyond matrices to
[[Linear map|linear operators]] on a [[Hilbert space]] <math>\mathcal H</math>.
<ref name="Redheffer60transform">
{{Cite journal|last=Redheffer|first=R. M.|
date=1960|title=On a Certain Linear Fractional Transformation|
url=https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm1960391269|
journal=Journal of Mathematics and Physics|language=en|volume=39|issue=1–4|
pages=269–286|doi=10.1002/sapm1960391269|issn=1467-9590}}
</ref>
.
By definition, <math>A_{ij}, B_{kl}</math> are linear endomorphisms of <math>\mathcal H</math>,
making <math>A, B</math> linear endomorphisms of <math>\mathcal H \oplus \mathcal H</math>,
where <math>\oplus</math> is the [[direct sum]].
However, the star product still makes sense as long as the transformations are compatible,
which is possible when <math>A \in \mathcal{L (H_\gamma \oplus H_\alpha, H_\alpha \oplus H_\gamma)}</math>
and <math>B \in \mathcal{L (H_\alpha \oplus H_\beta, H_\beta \oplus H_\alpha)}</math>
so that <math>A \star B \in \mathcal{L (H_\gamma \oplus H_\beta, H_\beta \oplus H_\gamma)}</math>.

== Properties ==

=== Existence ===
<math>(I - A_{12} B_{21})^{-1}</math> exists if and only if
<math>(I - B_{21} A_{12})^{-1}</math> exists.
<ref name="Mistiri86">
{{Cite journal|last=Mistiri|first=F.|date=1986-01-01|
title=The Star-product and its Algebraic Properties|
url=https://www.sciencedirect.com/science/article/abs/pii/0016003286900530|
journal=Journal of the Franklin Institute|language=en|volume=321|issue=1|
pages=21–38|doi=10.1016/0016-0032(86)90053-0|issn=0016-0032}}
</ref>
Thus when either exists, so does the Redheffer star product.

=== Identity ===
The star identity is the identity on <math>\mathcal H \oplus \mathcal H</math>,
or <math>\begin{pmatrix} I & 0 \\ 0 & I \end{pmatrix}</math>.
<ref name="Redheffer60transform" />

=== Associativity ===
The star product is associative, provided all of the relevant matrices are defined.
<ref name="Mistiri86" />
Thus <math>A \star B \star C = (A \star B) \star C = A \star (B \star C)</math>.

=== Adjoint ===
Provided either side exists, the [[Hermitian adjoint|adjoint]] of a Redheffer 
star product is <math>(A \star B)^* = B^* \star A^*</math>.
<ref name="Redheffer60transform" />

=== Inverse ===
If <math>B</math> is the left matrix inverse of <math>A</math> such that
<math>BA = I</math>, <math>A_{22}</math> has a right inverse, and
<math>A \star B</math> exists, then <math>A \star B = I</math>.
<ref name="Redheffer60transform" />
Similarly, if <math>B</math> is the left matrix inverse of <math>A</math> such
that <math>BA = I</math>, <math>A_{11}</math> has a right inverse, and
<math>B \star A</math> exists, then <math>B \star A = I</math>.

Also, if <math>A \star B = I</math> and <math>A_{22}</math> has a left inverse
then <math>BA = I</math>.

The star inverse equals the [[matrix inverse]] and both can be computed with
[[Block matrix#Block matrix operations|block inversion]] as
<ref name="Redheffer60transform " />

<math>\begin{pmatrix}
A_{11} & A_{12}
\\
A_{21} & A_{22}
\end{pmatrix}^{-1}
=
\begin{pmatrix}
(A_{11} - A_{12} A_{22}^{-1} A_{21})^{-1} & (A_{21} - A_{22} A_{12}^{-1} A_{11})^{-1}
\\
(A_{12} - A_{11} A_{21}^{-1} A_{22})^{-1} & (A_{22} - A_{21} A_{11}^{-1} A_{12})^{-1}
\end{pmatrix}</math>.

== Derivation from a linear system ==
[[File:Redheffer I-O.svg|thumb|The coupled system of equations, with arrows
labeling the inputs and outputs to each matrix]]
The star product arises from solving multiple linear systems of equations that share
variables in common.
Often, each linear system models the behavior of one subsystem in a physical process
and by connecting the multiple subsystems into a whole, one can eliminate variables
shared across subsystems in order to obtain the overall linear system.
For instance, let <math>\{ x_i \}_{i=1}^6</math> be elements of a [[Hilbert space]]
<math>\mathcal H</math> such that
<ref>
{{cite web |last1=Liu |first1=Victor |
title=On scattering matrices and the Redheffer star product |
url=http://victorliu.info/pdfs/Scombine.pdf |access-date=26 June 2021}}
</ref>

<math>\begin{pmatrix}
    x_3
    \\
    x_6
\end{pmatrix}
=
\begin{pmatrix}
    A_{11} & A_{12}
    \\
    A_{21} & A_{22}
\end{pmatrix}
\begin{pmatrix}
    x_5
    \\
    x_4
\end{pmatrix}</math>

and

<math>\begin{pmatrix}
    x_1
    \\
    x_4
\end{pmatrix}
=
\begin{pmatrix}
    B_{11} & B_{12}
    \\
    B_{21} & B_{22}
\end{pmatrix}
\begin{pmatrix}
    x_3
    \\
    x_2
\end{pmatrix}</math>
[[File:Redheffer system detail.svg|thumb|The "plumbing" of one of Redheffer's systems of equations.]]
giving the following <math>4</math> equations in <math>6</math> variables:

<math>\begin{align}
x_3 &= A_{11} x_5 + A_{12} x_4
\\
x_6 &= A_{21} x_5 + A_{22} x_4
\\
x_1 &= B_{11} x_3 + B_{12} x_2
\\
x_4 &= B_{21} x_3 + B_{22} x_2
\end{align}</math>.

By substituting the first equation into the last we find:

<math>x_4 = (I - B_{21}A_{12})^{-1} (B_{21}A_{11} x_5 + B_{22} x_2)</math>.

By substituting the last equation into the first we find:

<math>x_3 = (I - A_{12}B_{21})^{-1} (A_{11} x_5 + A_{12}B_{22} x_2)</math>.

Eliminating  <math>x_3, x_4</math>  by substituting the two preceding equations
into those for <math>x_1, x_6</math> results in the Redheffer star product
being the matrix such that:
<ref name="Redheffer59">
{{Cite journal|last=Redheffer|first=Raymond|date=1959|
title=Inequalities for a Matrix Riccati Equation|
url=https://www.jstor.org/stable/24900576|
journal=Journal of Mathematics and Mechanics|volume=8|issue=3|pages=349–367|jstor=24900576|
issn=0095-9057}}
</ref>

[[File:Redheffer star system.svg|thumb|The star product eliminates the shared
variables in this coupled system of equations.]]

<math>\begin{pmatrix}
    x_1
    \\
    x_6
\end{pmatrix}
= (A \star B)
\begin{pmatrix}
    x_5
    \\
    x_2
\end{pmatrix}</math>.

== Connection to scattering matrices ==
[[File:Scatter matrix detail.svg|thumb|The "plumbing" of the scattering matrix
has a different convention than Redheffer that amounts to swapping and relabeling
several quantities. The advantage is that now the S-matrix's subscripts label the
input and output ports as well as the block indices.]]
Many [[Scattering|scattering processes]] take on a form that motivates a different
convention for the block structure of the linear system of a scattering matrix.
Typically a physical device that performs a linear transformation on inputs, such as
linear dielectric media on electromagnetic waves or in quantum mechanical scattering,
can be encapsulated as a system which interacts with the environment through various
ports, each of which accepts inputs and returns outputs. It is conventional to use a different notation for the Hilbert space, <math>\mathcal H_i</math>, whose subscript
labels a port on the device.
Additionally, any element, <math>c_i^\pm \in \mathcal H_i</math>, has an additional superscript labeling the direction of travel (where + indicates moving from port i to i+1 and - indicates the reverse).

The equivalent notation for a Redheffer transformation,
<math>R \in \mathcal{L (H_1 \oplus H_2, H_2 \oplus H_1)}</math>,
used in the previous section is

<math>
\begin{pmatrix}
    c_2^+
    \\
    c_1^-
\end{pmatrix}
=
\begin{pmatrix}
    R_{11} & R_{12}
    \\
    R_{21} & R_{22}
\end{pmatrix}
\begin{pmatrix}
    c_1^+
    \\
    c_2^-
\end{pmatrix}
</math>
.

The action of the [[S-matrix]],
<math>S \in \mathcal{L (H_1 \oplus H_2, H_1 \oplus H_2)}</math>,
is defined with an additional flip compared to Redheffer's definition:<ref name="Rumpf11"/>

<math>
\begin{pmatrix}
    c_1^-
    \\
    c_2^+
\end{pmatrix}
=
\begin{pmatrix}
    S_{11} & S_{12}
    \\
    S_{21} & S_{22}
\end{pmatrix}
\begin{pmatrix}
    c_1^+
    \\
    c_2^-
\end{pmatrix}
</math>
,

so 
<math>
S =
\begin{pmatrix}
    0 & I
    \\
    I & 0
\end{pmatrix}
R
</math>
.
Note that for in order for the off-diagonal identity matrices to be defined,
we require <math>\mathcal{H_1, H_2}</math> be the same underlying Hilbert space.
(The subscript does not imply any difference, but is just a label for bookkeeping.)

The star product, <math>\star_S</math>,
for two S-matrices, <math>A, B</math>, is given by
<ref name="Rumpf11" />
[[File:Scatter connection.svg|thumb|The "plumbing" of a coupled pair of scattering matrices in a star product.]]
<math>
A \star_S B
=
\begin{pmatrix}
    A_{11} + A_{12} (I - B_{11} A_{22})^{-1} B_{11} A_{21} &
    A_{12} (I - B_{11} A_{22})^{-1} B_{12}
    \\
    B_{21} (I - A_{22} B_{11})^{-1} A_{21} &
    B_{22} + B_{21} (I - A_{22} B_{11})^{-1} A_{22} B_{12}
\end{pmatrix}
</math>
,

where <math>A \in \mathcal{L (H_1 \oplus H_2, H_1 \oplus H_2)}</math>
and <math>B \in \mathcal{L (H_2 \oplus H_3, H_2 \oplus H_3)}</math>,
so <math>A \star_S B \in \mathcal{L (H_1 \oplus H_3, H_1 \oplus H_3)}</math>.

=== Properties ===

These are analogues of the properties of <math>\star</math> for <math>\star_S</math>
Most of them follow from the correspondence
<math>J(A \star B) = (JA) \star_S (JB)</math>.
<math>J</math>, the exchange operator, is also the S-matrix star identity defined below.
For the rest of this section, <math>A,B,C</math> are S-matrices.

==== Existence ====

<math>A \star_S B</math> exists when either
<math>(I - A_{22} B_{11})^{-1}</math>
or
<math>(I - B_{11} A_{22})^{-1}</math>
exist.

==== Identity ====

The S-matrix star identity, <math>J</math>, is 
<math>
J =
\begin{pmatrix}
    0 & I
    \\
    I & 0
\end{pmatrix}
</math>.
This means <math>J \star_S S = S \star_S J = S</math>

==== Associativity ====

Associativity of <math>\star_S</math> follows from associativity of <math>\star</math> and of matrix multiplication.

==== Adjoint ====

From the correspondence between <math>\star</math> and <math>\star_S</math>,
and the adjoint of <math>\star</math>, we have that
<math>(A \star_S B)^* = J (B^* \star_S A^*) J</math>

==== Inverse ====

The matrix <math>\Sigma</math> that is the S-matrix star product inverse of
<math>S</math> in the sense that <math>\Sigma \star_S S = S \star_S \Sigma = J</math>
is <math>JS^{-1}J</math> where <math>S^{-1}</math> is the ordinary [[matrix inverse]]
and <math>J</math> is as defined above.

=== Connection to transfer matrices ===
[[File:Transfer detail.svg|thumb|Transfer matrices have a different "plumbing"
than scattering matrices. They connect one port to another instead of the inputs
at all ports to the outputs at all ports.]]
Observe that a scattering matrix can be rewritten as a 
[[Transfer-matrix method (optics)|transfer matrix]], <math>T</math>, with action
<math>\begin{pmatrix}
    c_2^+
    \\
    c_2^-
\end{pmatrix}
= T
\begin{pmatrix}
    c_1^+
    \\
    c_1^-
\end{pmatrix}</math>,
where
<ref name="Redheffer62" />

<math>
T =
\begin{pmatrix}
    T_{\scriptscriptstyle ++} & T_{\scriptscriptstyle +-}
    \\
    T_{\scriptscriptstyle -+} & T_{\scriptscriptstyle --}
\end{pmatrix}
=
\begin{pmatrix}
    S_{21} - S_{22} S_{12}^{-1} S_{11} & S_{22} S_{12}^{-1}
    \\
    - S_{12}^{-1} S_{11} & S_{12}^{-1}
\end{pmatrix}
</math>
.

Here the subscripts relate the different directions of propagation at each port.
As a result, the star product of scattering matrices

<math>
\begin{pmatrix}
    c_3^+
    \\
    c_1^-
\end{pmatrix}
= (S^A \star S^B)
\begin{pmatrix}
    c_1^+
    \\
    c_3^-
\end{pmatrix}
</math>
,

is analogous to the following matrix multiplication of transfer matrices
<ref name="Redheffer60supp" />

<math>
\begin{pmatrix}
    c_3^+
    \\
    c_3^-
\end{pmatrix}
= (T^A T^B)
\begin{pmatrix}
    c_1^+
    \\
    c_1^-
\end{pmatrix}
</math>
,

where <math>T^A \in \mathcal{L (H_1 \oplus H_1, H_2 \oplus H_2)}</math>
and <math>T^B \in \mathcal{L (H_2 \oplus H_2, H_3 \oplus H_3)}</math>,
so <math>T^A T^B \in \mathcal{L (H_1 \oplus H_1, H_3 \oplus H_3)}</math>.

== Generalizations ==
Redheffer generalized the star product in several ways:

=== Arbitrary bijections ===
If there is a bijection <math>M \leftrightarrow L</math> given by
<math>L = f(M)</math> then an associative star product can be defined by:
<ref name="Redheffer60supp">
{{Cite journal|last=Redheffer|first=Raymond|date=1960|
title=Supplementary Note on Matrix Riccati Equations|
url=https://www.jstor.org/stable/24900784|
journal=Journal of Mathematics and Mechanics|
volume=9|issue=5|pages=745–7f48|jstor=24900784|issn=0095-9057}}
</ref>

<math>A \star B = f^{-1} (f(A) f(B))</math>.

The particular star product defined by Redheffer above is obtained from:

<math>f(A) = ((I - A) + (I + A) J)^{-1} ((A - I) + (A + I) J)</math>

where <math>J(x, y) = (-x, y)</math>.

=== 3x3 star product ===
A star product can also be defined for 3x3 matrices.
<ref name="Redheffer60diffusion">
{{Cite journal|last=Redheffer|first=Raymond M.|date=1960|
title=The Mycielski-Paszkowski Diffusion Problem|
url=https://www.jstor.org/stable/24900958|
journal=Journal of Mathematics and Mechanics|volume=9|issue=4|pages=607–621|jstor=24900958|
issn=0095-9057}}
</ref>

== Applications to scattering matrices ==
In [[physics]], the Redheffer star product appears when constructing a total
[[scattering]] matrix from two or more subsystems.
If system <math>A</math> has a scattering matrix <math>S^A</math> and system
<math>B</math> has scattering matrix <math>S^B</math>, then the combined system
<math>AB</math> has scattering matrix <math>S^{AB} = S^A \star S^B</math>.
<ref name="Rumpf11">
{{Cite journal|last=Rumpf|first=Raymond C.|date=2011|
title=Improved Formulation of Scattering Matrices for Semi-Analytical Methods that is Consistent with Convention|
url=http://www.jpier.org/PIERB/pier.php?paper=11083107|
journal=Progress in Electromagnetics Research B|language=en|volume=35|
pages=241–261|doi=10.2528/PIERB11083107|issn=1937-6472|doi-access=free}}
</ref>

=== Transmission line theory ===
Many physical processes, including radiative transfer, neutron diffusion, circuit theory, and others are described by scattering processes whose formulation depends on the dimension of the process and the representation of the operators.<ref name="Redheffer62">
{{Cite journal|last=Redheffer|first=Raymond|date=1962|
title=On the Relation of Transmission-Line Theory to Scattering and Transfer|
url=https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm19624111|
journal=Journal of Mathematics and Physics|language=en|volume=41|issue=1–4|
pages=1–41|doi=10.1002/sapm19624111|issn=1467-9590}}</ref> For probabilistic problems, the scattering equation may appear in a [[Kolmogorov equations|Kolmogorov-type equation]].

=== Electromagnetism ===
The Redheffer star product can be used to solve for the propagation of electromagnetic fields in stratified, multilayered media.<ref name="Ko88">{{Cite journal|last1=Ko|first1=D. Y. K.|last2=Sambles|first2=J. R.|date=1988-11-01|
title=Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals|
url=https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-5-11-1863|
journal=JOSA A|language=EN|volume=5|issue=11|pages=1863–1866|
doi=10.1364/JOSAA.5.001863|bibcode=1988JOSAA...5.1863K|issn=1520-8532}}</ref> Each layer in the structure has its own scattering matrix and the total structure's scattering matrix can be described as the star product between all of the layers.<ref name="Whittaker99">
{{Cite journal|last1=Whittaker|first1=D. M.|last2=Culshaw|first2=I. S.|
date=1999-07-15|
title=Scattering-matrix treatment of patterned multilayer photonic structures|
url=https://link.aps.org/doi/10.1103/PhysRevB.60.2610|journal=Physical Review B|
volume=60|issue=4|pages=2610–2618|doi=10.1103/PhysRevB.60.2610|bibcode=1999PhRvB..60.2610W}}</ref> A free software program that simulates electromagnetism in layered media is the
[https://web.stanford.edu/group/fan/S4/  Stanford Stratified Structure Solver].

=== Semiconductor interfaces ===
Kinetic models of consecutive semiconductor interfaces can use a scattering matrix formulation to model the motion of electrons between the semiconductors.
<ref name="Gosse14">
{{Cite journal|last=Gosse|first=Laurent|date=2014-01-01|
title=Redheffer Products and Numerical Approximation of Currents in One-Dimensional Semiconductor Kinetic Models|
url=https://epubs.siam.org/doi/10.1137/130939584|
journal=Multiscale Modeling & Simulation|volume=12|issue=4|pages=1533–1560|
doi=10.1137/130939584|issn=1540-3459}}
</ref>

=== Factorization on graphs ===
In the analysis of Schrödinger operators on graphs, the scattering matrix of a graph can be obtained as a generalized star product of the scattering matrices corresponding to its subgraphs.<ref name="Kostrykin01">
{{Cite journal|last1=Kostrykin|first1=V.|last2=Schrader|first2=R.|date=2001-03-22|
title=The generalized star product and the factorization of scattering matrices on graphs|
url=https://aip.scitation.org/doi/10.1063/1.1354641|
journal=Journal of Mathematical Physics|volume=42|issue=4|pages=1563–1598|
doi=10.1063/1.1354641|arxiv=math-ph/0008022|bibcode=2001JMP....42.1563K|s2cid=6791638|issn=0022-2488}}
</ref>

== References ==
{{reflist}}

[[Category:Scattering theory|Scattering theory]]
[[Category:Scattering, absorption and radiative transfer (optics)|Scattering, absorption and radiative transfer]]
[[Category:Hilbert spaces]]
[[Category:Matrices]]
[[Category:Mathematical physics]]