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 {\displaystyle A,B} are the block matrices
A = ( A 11 A 12 A 21 A 22 ) {\displaystyle A={\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{pmatrix}}}
and
B = ( B 11 B 12 B 21 B 22 ) {\displaystyle B={\begin{pmatrix}B_{11}&B_{12}\\B_{21}&B_{22}\end{pmatrix}}} , whose blocks A i j , B k l {\displaystyle A_{ij},B_{kl}} have the same shape when
i j = k l {\displaystyle ij=kl} . The Redheffer star product is then defined by:
A ⋆ B = ( 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 ) {\displaystyle 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 ) {\displaystyle (I-A_{12}B_{21}),(I-B_{21}A_{12})} are invertible, where I {\displaystyle I} is an identity matrix conformable to A 12 B 21 {\displaystyle A_{12}B_{21}} or B 21 A 12 {\displaystyle 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 ) ⟺ A ( I − B A ) − 1 = ( I − A B ) − 1 A {\displaystyle (I-AB)A=A(I-BA)\iff A(I-BA)^{-1}=(I-AB)^{-1}A} . Redheffer's definition extends beyond matrices to linear operators on a Hilbert space H {\displaystyle {\mathcal {H}}} .
. By definition, A i j , B k l {\displaystyle A_{ij},B_{kl}} are linear endomorphisms of H {\displaystyle {\mathcal {H}}} , making A , B {\displaystyle A,B} linear endomorphisms of H ⊕ H {\displaystyle {\mathcal {H}}\oplus {\mathcal {H}}} , where ⊕ {\displaystyle \oplus } is the direct sum. However, the star product still makes sense as long as the transformations are compatible, which is possible when A ∈ L ( H γ ⊕ H α , H α ⊕ H γ ) {\displaystyle A\in {\mathcal {L(H_{\gamma }\oplus H_{\alpha },H_{\alpha }\oplus H_{\gamma })}}}
and B ∈ L ( H α ⊕ H β , H β ⊕ H α ) {\displaystyle B\in {\mathcal {L(H_{\alpha }\oplus H_{\beta },H_{\beta }\oplus H_{\alpha })}}}
so that A ⋆ B ∈ L ( H γ ⊕ H β , H β ⊕ H γ ) {\displaystyle A\star B\in {\mathcal {L(H_{\gamma }\oplus H_{\beta },H_{\beta }\oplus H_{\gamma })}}} .
Properties
Existence
( I − A 12 B 21 ) − 1 {\displaystyle (I-A_{12}B_{21})^{-1}} exists if and only if
( I − B 21 A 12 ) − 1 {\displaystyle (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 H ⊕ H {\displaystyle {\mathcal {H}}\oplus {\mathcal {H}}} , or ( I 0 0 I ) {\displaystyle {\begin{pmatrix}I&0\\0&I\end{pmatrix}}} .
Associativity The star product is associative, provided all of the relevant matrices are defined.
Thus A ⋆ B ⋆ C = ( A ⋆ B ) ⋆ C = A ⋆ ( B ⋆ C ) {\displaystyle 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 ⋆ B ) ∗ = B ∗ ⋆ A ∗ {\displaystyle (A\star B)^{*}=B^{*}\star A^{*}} .
Inverse If B {\displaystyle B} is the left matrix inverse of A {\displaystyle A} such that
B A = I {\displaystyle BA=I} , A 22 {\displaystyle A_{22}} has a right inverse, and
A ⋆ B {\displaystyle A\star B} exists, then A ⋆ B = I {\displaystyle A\star B=I} .
Similarly, if B {\displaystyle B} is the left matrix inverse of A {\displaystyle A} such that B A = I {\displaystyle BA=I} , A 11 {\displaystyle A_{11}} has a right inverse, and
B ⋆ A {\displaystyle B\star A} exists, then B ⋆ A = I {\displaystyle B\star A=I} . Also, if A ⋆ B = I {\displaystyle A\star B=I} and A 22 {\displaystyle A_{22}} has a left inverse then B A = I {\displaystyle BA=I} . The star inverse equals the matrix inverse and both can be computed with block inversion as
( A 11 A 12 A 21 A 22 ) − 1 = ( ( 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 ) {\displaystyle {\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 {\displaystyle \{x_{i}\}_{i=1}^{6}} be elements of a Hilbert space
H {\displaystyle {\mathcal {H}}} such that
( x 3 x 6 ) = ( A 11 A 12 A 21 A 22 ) ( x 5 x 4 ) {\displaystyle {\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
( x 1 x 4 ) = ( B 11 B 12 B 21 B 22 ) ( x 3 x 2 ) {\displaystyle {\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 {\displaystyle 4} equations in 6 {\displaystyle 6} variables:
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 {\displaystyle {\begin{aligned}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{aligned}}} . 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 ) {\displaystyle 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 ) {\displaystyle x_{3}=(I-A_{12}B_{21})^{-1}(A_{11}x_{5}+A_{12}B_{22}x_{2})} . Eliminating x 3 , x 4 {\displaystyle x_{3},x_{4}} by substituting the two preceding equations into those for x 1 , x 6 {\displaystyle x_{1},x_{6}} results in the Redheffer star product being the matrix such that:
( x 1 x 6 ) = ( A ⋆ B ) ( x 5 x 2 ) {\displaystyle {\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, H i {\displaystyle {\mathcal {H}}_{i}} , whose subscript labels a port on the device. Additionally, any element, c i ± ∈ H i {\displaystyle 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 ∈ L ( H 1 ⊕ H 2 , H 2 ⊕ H 1 ) {\displaystyle R\in {\mathcal {L(H_{1}\oplus H_{2},H_{2}\oplus H_{1})}}} , used in the previous section is
( c 2 + c 1 − ) = ( R 11 R 12 R 21 R 22 ) ( c 1 + c 2 − ) {\displaystyle {\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 ∈ L ( H 1 ⊕ H 2 , H 1 ⊕ H 2 ) {\displaystyle 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:
( c 1 − c 2 + ) = ( S 11 S 12 S 21 S 22 ) ( c 1 + c 2 − ) {\displaystyle {\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 = ( 0 I I 0 ) R {\displaystyle S={\begin{pmatrix}0&I\\I&0\end{pmatrix}}R}
. Note that in order for the off-diagonal identity matrices to be defined, we require H 1 , H 2 {\displaystyle {\mathcal {H_{1},H_{2}}}} be th