Redheffer star product explained

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 matricesA = \begin{pmatrix} A _{11} & A _{12} \\ A _{21} & A _{22} \end{pmatrix} andB = \begin{pmatrix} B _{11} & B _{12} \\ B _{21} & B _{22} \end{pmatrix} ,whose blocks A _{ij} , B _{kl} have the same shape whenij = 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 conformableto A _{12} B _{21} or B _{21} A _{12} , respectively.This can be rewritten several ways making use of the so-calledpush-through identity(I - A B) A = A (I - B A) \iff A (I - B A) ^{-1} = (I - A B) ^{-1} A .l H .^[1] .By definition, A _{ij} , B _{kl} are linear endomorphisms of l H ,making A, B linear endomorphisms of l H ⊕ l H ,where ⊕ is the direct sum.However, the star product still makes sense as long as the transformations are compatible,which is possible when A \in l{L (H _{\gamma} ⊕ H _{\alpha,} H _{\alpha} ⊕ H _{\gamma)}} and B \in l{L (H _{\alpha} ⊕ H _{\beta,} H _{\beta} ⊕ H _{\alpha)}} so that A \star B \in l{L (H _{\gamma} ⊕ H _{\beta,} H _{\beta} ⊕ H _{\gamma)}} .

Properties

Existence

(I - A _{12} B _{21} ) ^{-1} exists if and only if(I - B _{21} A _{12} ) ^{-1} exists.^[2] Thus when either exists, so does the Redheffer star product.

Identity

The star identity is the identity on

l H ⊕ l 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 thatBA = I , A _{22} has a right inverse, andA \star B exists, then A \star B = I .Similarly, if B is the left matrix inverse of A suchthat BA = I , A _{11} has a right inverse, andB \star A exists, then B \star A = I .

Also, if

A \star B = I and A _{22} has a left inversethen BA = I .

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

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

Derivation from a linear system

The star product arises from solving multiple linear systems of equations that sharevariables in common.Often, each linear system models the behavior of one subsystem in a physical processand by connecting the multiple subsystems into a whole, one can eliminate variablesshared across subsystems in order to obtain the overall linear system.For instance, let

\{ x _{i} \} ^{6} _{i =1} be elements of a Hilbert spacel H such that^[3] \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 equationsinto those for x _{1,} x _{6} results in the Redheffer star productbeing the matrix such that:^[4] \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 differentconvention for the block structure of the linear system of a scattering matrix.Typically a physical device that performs a linear transformation on inputs, such aslinear dielectric media on electromagnetic waves or in quantum mechanical scattering,can be encapsulated as a system which interacts with the environment through variousports, each of which accepts inputs and returns outputs. It is conventional to use a different notation for the Hilbert space,

l H _{i} , whose subscriptlabels a port on the device.Additionally, any element, c ^{\pm} _{i} \in l 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 l{L (H _{1} ⊕ H _{2,} H _{2} ⊕ H _{1)}} ,used in the previous section is\begin{pmatrix} c ^{+} _{2} \\ c ^{- \end{pmatrix} = \begin{pmatrix}} _{1} R _{11} & R _{12} \\ R _{21} & R _{22} \end{pmatrix} \begin{pmatrix} c ^{+} _{1} \\ c ^{- \end{pmatrix}} _{2} .

The action of the S-matrix,

S \in l{L (H _{1} ⊕ H _{2,} H _{1} ⊕ H _{2)}} ,is defined with an additional flip compared to Redheffer's definition:\begin{pmatrix} c ^{-} _{1} \\ c ^{+ \end{pmatrix} = \begin{pmatrix}} _{2} S _{11} & S _{12} \\ S _{21} & S _{22} \end{pmatrix} \begin{pmatrix} c ^{+} _{1} \\ c ^{- \end{pmatrix}} _{2} ,

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 l{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 byA \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 l{L (H _{1} ⊕ H _{2,} H _{1} ⊕ H _{2)}} and B \in l{L (H _{2} ⊕ H _{3,} H _{2} ⊕ H _{3)}} ,so A \star _{S} B \in l{L (H _{1} ⊕ H _{3,} H _{1} ⊕ H _{3)}} .

Properties

These are analogues of the properties of

\star for \star _{S} Most of them follow from the correspondenceJ(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 ofS in the sense that \Sigma \star _{S} S = S \star _{S} \Sigma = J is JS ^{-1} J where S ^{-1} is the ordinary matrix inverseand 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 ^{- \end{pmatrix} =} _{2} T \begin{pmatrix} c ^{+} _{1} \\ c ^{- \end{pmatrix}} _{1} ,whereT = \begin{pmatrix} T _{\scriptscriptstyle} & T _{\scriptscriptstyle} \\ T _{\scriptscriptstyle} & T _{\scriptscriptstyle} \end{pmatrix} = \begin{pmatrix} S _{21} - S _{22} S ^{-1} _{12} S _{11} & S _{22} S ^{-1} _{12} \\ - S ^{-1} _{12} S _{11} & S ^{-1} _{12} \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 ^{- \end{pmatrix} =} _{1} (S ^{A} \star S ^{B) \begin{pmatrix}} c ^{+} _{1} \\ c ^{- \end{pmatrix}} _{3} ,

is analogous to the following matrix multiplication of transfer matrices

\begin{pmatrix} c ^{+} _{3} \\ c ^{- \end{pmatrix} =} _{3} (T ^{A} T ^{B) \begin{pmatrix}} c ^{+} _{1} \\ c ^{- \end{pmatrix}} _{1} ,

where

T ^{A} \in l{L (H _{1} ⊕ H _{1,} H _{2} ⊕ H _{2)}} and T ^{B} \in l{L (H _{2} ⊕ H _{2,} H _{3} ⊕ H _{3)}} ,so T ^{A} T ^{B} \in l{L (H _{1} ⊕ H _{1,} H _{3} ⊕ H _{3)}} .

Generalizations

Redheffer generalized the star product in several ways:

Arbitrary bijections

If there is a bijection

M \leftrightarrow L given byL = f(M) then an associative star product can be defined by:^[5] 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.^[6]

Applications to scattering matrices

In physics, the Redheffer star product appears when constructing a totalscattering matrix from two or more subsystems.If system

A has a scattering matrix S ^{A} and systemB has scattering matrix S ^{B} , then the combined systemAB has scattering matrix S ^{AB} = S ^{A} \star S ^{B} .^[7]

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.^[8] 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.^[9] 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.^[10] 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.^[11]

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.^[12]

Notes and References

1. Redheffer. R. M.. 1960. On a Certain Linear Fractional Transformation. Journal of Mathematics and Physics. en. 39. 1–4. 269–286. 10.1002/sapm1960391269. 1467-9590.
2. Mistiri. F.. 1986-01-01. The Star-product and its Algebraic Properties. Journal of the Franklin Institute. en. 321. 1. 21–38. 10.1016/0016-0032(86)90053-0. 0016-0032.
3. Web site: Liu . Victor . On scattering matrices and the Redheffer star product . 26 June 2021.
4. Redheffer. Raymond. 1959. Inequalities for a Matrix Riccati Equation. Journal of Mathematics and Mechanics. 8. 3. 349–367. 24900576. 0095-9057.
5. Redheffer. Raymond. 1960. Supplementary Note on Matrix Riccati Equations. Journal of Mathematics and Mechanics. 9. 5. 745–7f48. 24900784. 0095-9057.
6. Redheffer. Raymond M.. 1960. The Mycielski-Paszkowski Diffusion Problem. Journal of Mathematics and Mechanics. 9. 4. 607–621. 24900958. 0095-9057.
7. Rumpf. Raymond C.. 2011. Improved Formulation of Scattering Matrices for Semi-Analytical Methods that is Consistent with Convention. Progress in Electromagnetics Research B. en. 35. 241–261. 10.2528/PIERB11083107. 1937-6472. free.
8. Redheffer. Raymond. 1962. On the Relation of Transmission-Line Theory to Scattering and Transfer. Journal of Mathematics and Physics. en. 41. 1–4. 1–41. 10.1002/sapm19624111. 1467-9590.
9. Ko. D. Y. K.. Sambles. J. R.. 1988-11-01. Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals. JOSA A. EN. 5. 11. 1863–1866. 10.1364/JOSAA.5.001863. 1988JOSAA...5.1863K. 1520-8532.
10. Whittaker. D. M.. Culshaw. I. S.. 1999-07-15. Scattering-matrix treatment of patterned multilayer photonic structures. Physical Review B. 60. 4. 2610–2618. 10.1103/PhysRevB.60.2610. 1999PhRvB..60.2610W.
11. Gosse. Laurent. 2014-01-01. Redheffer Products and Numerical Approximation of Currents in One-Dimensional Semiconductor Kinetic Models. Multiscale Modeling & Simulation. 12. 4. 1533–1560. 10.1137/130939584. 1540-3459.
12. Kostrykin. V.. Schrader. R.. 2001-03-22. The generalized star product and the factorization of scattering matrices on graphs. Journal of Mathematical Physics. 42. 4. 1563–1598. 10.1063/1.1354641. math-ph/0008022. 2001JMP....42.1563K. 6791638. 0022-2488.