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,[1] 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

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Suppose   are the block matrices   and  , whose blocks   have the same shape when  . The Redheffer star product is then defined by: [1]

  ,

assuming that   are invertible, where   is an identity matrix conformable to   or  , respectively. This can be rewritten several ways making use of the so-called push-through identity  .

Redheffer's definition extends beyond matrices to linear operators on a Hilbert space  . [2] . By definition,   are linear endomorphisms of  , making   linear endomorphisms of  , where   is the direct sum. However, the star product still makes sense as long as the transformations are compatible, which is possible when   and   so that  .

Properties

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Existence

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  exists if and only if   exists. [3] Thus when either exists, so does the Redheffer star product.

Identity

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The star identity is the identity on  , or  . [2]

Associativity

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The star product is associative, provided all of the relevant matrices are defined. [3] Thus  .

Adjoint

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Provided either side exists, the adjoint of a Redheffer star product is  . [2]

Inverse

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If   is the left matrix inverse of   such that  ,   has a right inverse, and   exists, then  . [2] Similarly, if   is the left matrix inverse of   such that  ,   has a right inverse, and   exists, then  .

Also, if   and   has a left inverse then  .

The star inverse equals the matrix inverse and both can be computed with block inversion as [2]

 .

Derivation from a linear system

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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   be elements of a Hilbert space   such that [4]

 

and

 

 
The "plumbing" of one of Redheffer's systems of equations.

giving the following   equations in   variables:

 .

By substituting the first equation into the last we find:

 .

By substituting the last equation into the first we find:

 .

Eliminating   by substituting the two preceding equations into those for   results in the Redheffer star product being the matrix such that: [1]

 
The star product eliminates the shared variables in this coupled system of equations.

 .

Connection to scattering matrices

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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 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,  , whose subscript labels a port on the device. Additionally, any element,  , 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,  , used in the previous section is

  .

The action of the S-matrix,  , is defined with an additional flip compared to Redheffer's definition:[5]

  ,

so   . Note that for in order for the off-diagonal identity matrices to be defined, we require   be the same underlying Hilbert space. (The subscript does not imply any difference, but is just a label for bookkeeping.)

The star product,  , for two S-matrices,  , is given by [5]

 
The "plumbing" of a coupled pair of scattering matrices in a star product.

  ,

where   and  , so  .

Properties

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These are analogues of the properties of   for   Most of them follow from the correspondence  .  , the exchange operator, is also the S-matrix star identity defined below. For the rest of this section,   are S-matrices.

Existence

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  exists when either   or   exist.

Identity

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The S-matrix star identity,  , is  . This means  

Associativity

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Associativity of   follows from associativity of   and of matrix multiplication.

Adjoint

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From the correspondence between   and  , and the adjoint of  , we have that  

Inverse

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The matrix   that is the S-matrix star product inverse of   in the sense that   is   where   is the ordinary matrix inverse and   is as defined above.

Connection to transfer matrices

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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,  , with action  , where [6]

  .

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

  ,

is analogous to the following matrix multiplication of transfer matrices [7]

  ,

where   and  , so  .

Generalizations

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Redheffer generalized the star product in several ways:

Arbitrary bijections

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If there is a bijection   given by   then an associative star product can be defined by: [7]

 .

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

 

where  .

3x3 star product

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A star product can also be defined for 3x3 matrices. [8]

Applications to scattering matrices

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In physics, the Redheffer star product appears when constructing a total scattering matrix from two or more subsystems. If system   has a scattering matrix   and system   has scattering matrix  , then the combined system   has scattering matrix  . [5]

Transmission line theory

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

Electromagnetism

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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

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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

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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]

References

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  1. ^ a b c Redheffer, Raymond (1959). "Inequalities for a Matrix Riccati Equation". Journal of Mathematics and Mechanics. 8 (3): 349–367. ISSN 0095-9057. JSTOR 24900576.
  2. ^ a b c d e Redheffer, R. M. (1960). "On a Certain Linear Fractional Transformation". Journal of Mathematics and Physics. 39 (1–4): 269–286. doi:10.1002/sapm1960391269. ISSN 1467-9590.
  3. ^ a b Mistiri, F. (1986-01-01). "The Star-product and its Algebraic Properties". Journal of the Franklin Institute. 321 (1): 21–38. doi:10.1016/0016-0032(86)90053-0. ISSN 0016-0032.
  4. ^ Liu, Victor. "On scattering matrices and the Redheffer star product" (PDF). Retrieved 26 June 2021.
  5. ^ a b c Rumpf, Raymond C. (2011). "Improved Formulation of Scattering Matrices for Semi-Analytical Methods that is Consistent with Convention". Progress in Electromagnetics Research B. 35: 241–261. doi:10.2528/PIERB11083107. ISSN 1937-6472.
  6. ^ a b Redheffer, Raymond (1962). "On the Relation of Transmission-Line Theory to Scattering and Transfer". Journal of Mathematics and Physics. 41 (1–4): 1–41. doi:10.1002/sapm19624111. ISSN 1467-9590.
  7. ^ a b Redheffer, Raymond (1960). "Supplementary Note on Matrix Riccati Equations". Journal of Mathematics and Mechanics. 9 (5): 745–7f48. ISSN 0095-9057. JSTOR 24900784.
  8. ^ Redheffer, Raymond M. (1960). "The Mycielski-Paszkowski Diffusion Problem". Journal of Mathematics and Mechanics. 9 (4): 607–621. ISSN 0095-9057. JSTOR 24900958.
  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. 5 (11): 1863–1866. Bibcode:1988JOSAA...5.1863K. doi:10.1364/JOSAA.5.001863. ISSN 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. Bibcode:1999PhRvB..60.2610W. doi:10.1103/PhysRevB.60.2610.
  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. doi:10.1137/130939584. ISSN 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. arXiv:math-ph/0008022. Bibcode:2001JMP....42.1563K. doi:10.1063/1.1354641. ISSN 0022-2488. S2CID 6791638.