Redheffer star product

In mathematics, the Redheffer star product is a binary operation on linear operators that arises in connection to solving coupled systems of linear equations. It was introduced by Raymond Redheffer in 1959, and has subsequently been widely adopted in computational methods for 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 channels of one scatterer are connected to inputs of another scatterer.

Definition
Suppose $$A, B$$ are the block matrices $$A = \begin{pmatrix} A_{11} & A_{12} \\   A_{21} & A_{22} \end{pmatrix}$$ and $$B = \begin{pmatrix} B_{11} & B_{12} \\   B_{21} & B_{22} \end{pmatrix} $$, whose blocks $$A_{ij}, B_{kl}$$ have the same shape when $$ij = kl$$. The Redheffer star product is then defined by:

$$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} $$ ,

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

Redheffer's definition extends beyond matrices to linear operators on a Hilbert space $$\mathcal H$$. . By definition, $$A_{ij}, B_{kl}$$ are linear endomorphisms of $$\mathcal H$$, making $$A, B$$ linear endomorphisms of $$\mathcal H \oplus \mathcal H$$, where $$\oplus$$ is the direct sum. However, the star product still makes sense as long as the transformations are compatible, which is possible when $$A \in \mathcal{L (H_\gamma \oplus H_\alpha, H_\alpha \oplus H_\gamma)}$$ and $$B \in \mathcal{L (H_\alpha \oplus H_\beta, H_\beta \oplus H_\alpha)}$$ so that $$A \star B \in \mathcal{L (H_\gamma \oplus H_\beta, H_\beta \oplus H_\gamma)}$$.

Existence
$$(I - A_{12} B_{21})^{-1}$$ exists if and only if $$(I - B_{21} A_{12})^{-1}$$ exists. Thus when either exists, so does the Redheffer star product.

Identity
The star identity is the identity on $$\mathcal H \oplus \mathcal H$$, or $$\begin{pmatrix} I & 0 \\ 0 & I \end{pmatrix}$$.

Associativity
The star product is associative, provided all of the relevant matrices are defined. Thus $$A \star B \star C = (A \star B) \star C = A \star (B \star C)$$.

Adjoint
Provided either side exists, the adjoint of a Redheffer star product is $$(A \star B)^* = B^* \star A^*$$.

Inverse
If $$B$$ is the left matrix inverse of $$A$$ such that $$BA = I$$, $$A_{22}$$ has a right inverse, and $$A \star B$$ exists, then $$A \star B = I$$. Similarly, if $$B$$ is the left matrix inverse of $$A$$ such that $$BA = I$$, $$A_{11}$$ has a right inverse, and $$B \star A$$ exists, then $$B \star A = I$$.

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

The star inverse equals the matrix inverse and both can be computed with block inversion as

$$\begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}^{-1} = \begin{pmatrix} (A_{11} - A_{12} A_{22}^{-1} A_{21})^{-1} & (A_{21} - A_{22} A_{12}^{-1} A_{11})^{-1} \\ (A_{12} - A_{11} A_{21}^{-1} A_{22})^{-1} & (A_{22} - A_{21} A_{11}^{-1} A_{12})^{-1} \end{pmatrix}$$.

Derivation from a linear system
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 $$\{ x_i \}_{i=1}^6$$ be elements of a Hilbert space $$\mathcal H$$ such that

$$\begin{pmatrix} x_3 \\   x_6 \end{pmatrix} = \begin{pmatrix} A_{11} & A_{12} \\   A_{21} & A_{22} \end{pmatrix} \begin{pmatrix} x_5 \\   x_4 \end{pmatrix}$$

and

$$\begin{pmatrix} x_1 \\   x_4 \end{pmatrix} = \begin{pmatrix} B_{11} & B_{12} \\   B_{21} & B_{22} \end{pmatrix} \begin{pmatrix} x_3 \\   x_2 \end{pmatrix}$$ giving the following $$4$$ equations in $$6$$ variables:

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

By substituting the first equation into the last we find:

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

By substituting the last equation into the first we find:

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

Eliminating $$x_3, x_4$$  by substituting the two preceding equations into those for $$x_1, x_6$$ results in the Redheffer star product being the matrix such that:



$$\begin{pmatrix} x_1 \\   x_6 \end{pmatrix} = (A \star B) \begin{pmatrix} x_5 \\   x_2 \end{pmatrix}$$.

Connection to scattering matrices
Many 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, $$\mathcal H_i$$, whose subscript labels a port on the device. Additionally, any element, $$c_i^\pm \in \mathcal H_i$$, 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, $$R \in \mathcal{L (H_1 \oplus H_2, H_2 \oplus H_1)}$$, used in the previous section is

$$ \begin{pmatrix} c_2^+ \\   c_1^- \end{pmatrix} = \begin{pmatrix} R_{11} & R_{12} \\   R_{21} & R_{22} \end{pmatrix} \begin{pmatrix} c_1^+ \\   c_2^- \end{pmatrix} $$.

The action of the S-matrix, $$S \in \mathcal{L (H_1 \oplus H_2, H_1 \oplus H_2)}$$, is defined with an additional flip compared to Redheffer's definition:

$$ \begin{pmatrix} c_1^- \\   c_2^+ \end{pmatrix} = \begin{pmatrix} S_{11} & S_{12} \\   S_{21} & S_{22} \end{pmatrix} \begin{pmatrix} c_1^+ \\   c_2^- \end{pmatrix} $$ ,

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

The star product, $$\star_S$$, for two S-matrices, $$A, B$$, is given by $$ A \star_S B = \begin{pmatrix} A_{11} + A_{12} (I - B_{11} A_{22})^{-1} B_{11} A_{21} & A_{12} (I - B_{11} A_{22})^{-1} B_{12} \\   B_{21} (I - A_{22} B_{11})^{-1} A_{21} & B_{22} + B_{21} (I - A_{22} B_{11})^{-1} A_{22} B_{12} \end{pmatrix} $$ ,

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

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

Existence
$$A \star_S B$$ exists when either $$(I - A_{22} B_{11})^{-1}$$ or $$(I - B_{11} A_{22})^{-1}$$ exist.

Identity
The S-matrix star identity, $$J$$, is $$ J = \begin{pmatrix} 0 & I   \\ I & 0 \end{pmatrix} $$. This means $$J \star_S S = S \star_S J = S$$

Associativity
Associativity of $$\star_S$$ follows from associativity of $$\star$$ and of matrix multiplication.

Adjoint
From the correspondence between $$\star$$ and $$\star_S$$, and the adjoint of $$\star$$, we have that $$(A \star_S B)^* = J (B^* \star_S A^*) J$$

Inverse
The matrix $$\Sigma$$ that is the S-matrix star product inverse of $$S$$ in the sense that $$\Sigma \star_S S = S \star_S \Sigma = J$$ is $$JS^{-1}J$$ where $$S^{-1}$$ is the ordinary matrix inverse and $$J$$ is as defined above.

Connection to transfer matrices
Observe that a scattering matrix can be rewritten as a transfer matrix, $$T$$, with action $$\begin{pmatrix} c_2^+ \\   c_2^- \end{pmatrix} = T \begin{pmatrix} c_1^+ \\   c_1^- \end{pmatrix}$$, where

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

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

$$ \begin{pmatrix} c_3^+ \\   c_1^- \end{pmatrix} = (S^A \star S^B) \begin{pmatrix} c_1^+ \\   c_3^- \end{pmatrix} $$ ,

is analogous to the following matrix multiplication of transfer matrices

$$ \begin{pmatrix} c_3^+ \\   c_3^- \end{pmatrix} = (T^A T^B) \begin{pmatrix} c_1^+ \\   c_1^- \end{pmatrix} $$ ,

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

Generalizations
Redheffer generalized the star product in several ways:

Arbitrary bijections
If there is a bijection $$M \leftrightarrow L$$ given by $$L = f(M)$$ then an associative star product can be defined by:

$$A \star B = f^{-1} (f(A) f(B))$$.

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

$$f(A) = ((I - A) + (I + A) J)^{-1} ((A - I) + (A + I) J)$$

where $$J(x, y) = (-x, y)$$.

3x3 star product
A star product can also be defined for 3x3 matrices.

Applications to scattering matrices
In physics, the Redheffer star product appears when constructing a total scattering matrix from two or more subsystems. If system $$A$$ has a scattering matrix $$S^A$$ and system $$B$$ has scattering matrix $$S^B$$, then the combined system $$AB$$ has scattering matrix $$S^{AB} = S^A \star S^B$$.

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. For probabilistic problems, the scattering equation may appear in a Kolmogorov-type equation.

Electromagnetism
The Redheffer star product can be used to solve for the propagation of electromagnetic fields in stratified, multilayered media. 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. A free software program that simulates electromagnetism in layered media is the 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.

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.