aa r X i v : . [ m a t h . R A ] J a n Matrix constructs ∗ Ted Hurley † & Barry Hurley ‡ Abstract
Matrices can be built and designed by applying procedures from lower order matrices. Matrix tensorproducts, direct sums of matrices and multiplication of matrices retain certain properties of the lowerorder matrices; matrices produced by these procedures are said to be separable . Here builders of ma-trices which retain properties of the lower order matrices or acquire new properties are described; theconstructions are not separable. The builders may be combined to construct series of matrix types.A number of applications are given. The new systems allow the design of multidimensional non-separable systems. Methods with which to design multidimensional paraunitary matrices are derived;these have applications for wavelet and filter bank design. New entangled unitary matrices may bedesigned; these may be used in quantum information theory. Full diversity constellations of unitarymatrices for space time applications are efficiently designed by the constructions.
Matrices may be built and designed using matrix procedures which ideally should retain properties of theconstituents or acquire new properties. For example a direct sum or tensor product of invertible matricesis invertible, a tensor product or direct sum of unitary matrices is a unitary matrix. A matrix built usinga matrix tensor product, a matrix direct sum or a matrix multiplication is said to be a separable matrix;constructions which are non-separable (entangled) may be required. Here building blocks for matrices whichretain properties of the builders or acquire new properties, not inherent in the constituents, are presented.In general the matrices acquired are non-separable. The constructions enable series of matrices of requiredtypes to be built, such as unitary or paraunitary matrices, or particular types of unitary or paraunitarymatrices and others.Designs and applications occur naturally. Building blocks for paraunitary matrices are fundamental insignal processing. Filter banks play an important role in signal processing but multidimensional filter bankshave been hard to design. In the huge research area of multirate filterbanks and wavelets, paraunitarymatrices play a fundamental role, [3, 11]; see Section 2.6 for further background. Full diversity sets ofconstellations of unitary matrices of many forms and of good quality are designed from the constructions,section 3; see [12] and [6] for background on such constellation requirements. Non-separable unitary matriceshave applications in diverse areas such as quantum information theory, [14].Non-separable multidimensional systems are designed and these can capture geometric structure ratherthan those constructed from one dimensional schemes using separable constructs.Infinite series of matrices can be built and the design process may be selected at each stage. Applicationsin cryptography are immediate.The basic types of constructions are described separately in subsections 2.1 and 2.2 and then are combinedsubsection 2.8. Systems including unitary matrices, paraunitary matrices, constellations of unitary matricesand types of these are designed from these basic schemes separately and then in conjunction with one another. ∗ MSC 2020 Classification:15A30, 16S50, 05B20, 94A15 † National Universiy of Ireland Galway, email: [email protected] ‡ Friar’s Hill, Galway, email: Barryj [email protected]
Basic algebra notation and background may be found in many books on matrix theory or linear algebra andalso online. In general matrices are formed over any ring including polynomial rings. A T denotes the transpose of the matrix A . For a matrix A over C , A ∗ denotes the complex conjugatetransposed; over other rings by convention A ∗ = A T . Now I n denotes the identity n × n matrix which is alsodenoted by I when the size is understood. Also 1 is used for the identity of the ring under consideration; itmay be used for I or I n as appropriate. Say A is a unitary n × n matrix provided AA ∗ = I n and say H is asymmetric (often called Hermitian) matrix provided H ∗ = H . A one-dimensional (1D) paraunitary matrixover C is a square matrix U ( z ) satisfying U ( z ) U ∗ ( z − ) = 1. In general a k -dimensional (kD) paraunitarymatrix over C is a matrix U ( z ), where z = ( z , z , . . . , z k ) is a vector of (commuting) variables { z , z , . . . , z k } ,such that U ( z ) U ∗ ( z − ) = 1 with the definition z − = ( z − , z − , . . . , z − k ). Over fields other than C aparaunitary matrix is a matrix U ( z ) satisfying U ( z ) U T ( z − ) = 1.An idempotent matrix E is a matrix satisfying E = E . The idempotent is symmetric provided E ∗ = E ;idempotents here are all symmetric. A complete orthogonal symmetric idempotent (COSI) set of n × n idempotents is a set { E , E , . . . , E k } where each E i is a symmetric idempotent, E i E j = 0 = E j E i for i = j and E + E + . . . + E k = I n . Further definitions are given as necessary within sections. Definitions relatedto constellations of unitary matrices are given in Section 3; definitions related to Hadamard matrices, realand complex, are given in section 2.7. Methods are developed to design and construct types of matrices using complete orthogonal symmetricidempotent (COSI) sets. These are non-separable structures and are different to the constructions describedin section 2.2 below. Using COSI sets for constructing unitary and paraunitary matrices was initiated in [7].
Proposition 2.1
Let { E , E , . . . , E k } be a COSI set in C n . Define G = E E . . . E k E E . . . E k ... ... ... ... E k E k . . . E kk whereeach E j appears once in each (block) row and once in each (block) column. Then G is a unitary matrix. Proof:
Take the block inner product of two different rows of blocks. The E i are orthogonal to one anotherso the result is 0. Take the block inner product of the row, j , of blocks with itself. This gives E j + E j + . . . + E jk = E j + E j + . . . + E jk = 1(= I n ). Hence GG ∗ = 1(= I nk ). (cid:3) A block circulant matrix is one of the form A A . . . A n A n A . . . A n − ... ... ... ... A A . . . A where the A i are blocks of the same2ize. A reverse circulant block matrix is one of the form A A . . . A n − A n A A . . . A n A ... ... ... ... ... A n A . . . A n − A n − where the A i areblocks of the same size. A circulant block matrix may be transformed into a reverse circulant block matrixby block row operations.For example E E E E E E E E E E E E E E E E ←→ E E E E E E E E E E E E E E E E , where ←→ here indicates that one canbe obtained from the other by block row operations. The one on the left is block circulant and the one onthe right is reverse block circulant.In particular given a COSI set { E , E , . . . , E k } , block circulant unitary matrices and block reversecirculant unitary matrices may be formed. Note that the block reverse circulant matrix is symmetric as the E i are symmetric.When variables are attached to the E i a paraunitary matrix is obtained; when elements of modulus 1are attached to the E i a unitary matrix is obtained. Proposition 2.2
Let { E , E , . . . , E k } be a COSI set in C n and define G = E α E α . . . E k α k E α E α . . . E k α k ... ... ... ... E k α k E k α k . . . E kk α kk where each E k appears once in each (block) row and once in each (block) column.(i) Let the α ij be variables. Then GG ∗ = I and G is a paraunitary matrix. (For a variable α , α ∗ = α − .)(ii) If | α ij | = 1 for each α ij , then G is a unitary matrix. The proof is similar to the proof of Proposition 2.1.Block circulant and block reverse circulant matrices may be formed. The reverse circulant block matrixis symmetric provided α ij = α ∗ ji .Large constructions and designs may be formed efficiently on computer. Example 2.1
Let E = ( ) , E = (cid:0) − − (cid:1) . Then { E , E } is a COSI set. Define W = (cid:16) xE yE zE tE (cid:17) = x x y − yx x − y yz − z t t − z z t t . Then W W ∗ = I .Let x = 1 = t = y = z in W and the following matrix is obtained. H = (cid:18) −
11 1 − − − (cid:19) ; this is acommon matrix used, or given as an example, in quantum theory as non-separable. Non-separability of aparaunitary matrix is a strong condition. Example 2.2 Q = (cid:0) i − i (cid:1) , Q = (cid:0) − ii (cid:1) . Then { Q , Q } is a COSI set.Let Q = (cid:16) xQ yQ zQ tQ (cid:17) .Then Q is a paraunitary matrix. Now letting the variables have complex values of modulus gives riseto complex Hadamard matrices as for example (cid:18) i − i − i i − i ii − i (cid:19) . xample 2.3 Return to the matrices in Example 2.1 with E = (cid:0) (cid:1) , E = (cid:0) − − (cid:1) . Then G = (cid:18) E E E E (cid:19) and H = (cid:18) E E E E (cid:19) are unitary matrices. Then × G, × H are Hadamard real × matrices.Take the columns { u , u , u , u } of G and form F i = u i u ∗ i so that { F , F , F , F } is a COSI set. Thesemay then be used to form × unitary matrices; the entries are ± and thus times these matrices areHadamard × matrices. In particular G = F F F F F F F F F F F F F F F F is a symmetric unitary matrix and thus × G is a symmetric Hadamard real matrix. There are many ways and directions with which to design and construct COSI sets. We list here someproperties of idempotents which are well-known or easily deduced. • Let { u , u , . . . , u k } be an orthonormal set of column vectors; then { E , E , . . . , E k } is an orthogonalsymmetric idempotent set where E i = u i u ∗ i . If S = { E , E , . . . , E k } is not complete, set E =( I − E − E − . . . − E k ) and then { E , E , . . . , E k , E } is a COSI set. • If { E , E , . . . , E k } is an orthogonal symmetric idempotent set, then rank( P ki =1 E i ) = P ki =1 (rank E i ). • If E is an idempotent of rank k then E is the sum of k orthogonal idempotents of rank 1. A methodfor writing such an idempotent as the sum of rank 1 idempotents is given in [13]. • When U is unitary, its columns { u , u , . . . , u n } form an orthonormal basis and thus { E , E , . . . , E n } with E i = u i u ∗ i is a complete orthogonal symmetric idempotent set which may then be used to formunitary or paraunitary matrices. • If { E, F } are orthogonal idempotents then E + F is an idempotent orthogonal to any idempotentwhich is orthogonal to both E, F . Thus if { E, F, K , K , . . . , K t } is an orthogonal idempotent set sois E + F, K , K , . . . , K t and if { E, F, K , K . . . , K t } is a COSI set so is { E + F, K , . . . , K t } Orthogonal idempotents may be combined to form new idempotents; elements in a COSI set may becombined to form a new COSI set. This new COSI set may then be used to design unitary, paraunitarymatrices and others. The following examples illustrate the general method, which is very useful. Denote thecirculant matrix a a . . . a k a k a . . . a ... ... ... ... a a . . . a by circ( a , a , . . . , a k ). Note also that if ω = e iθ then ω + ω ∗ = 2 cos θ . Example 2.4
Denote the columns of the × normalised Fourier matrix by { u , u , u , u , u } . Define E i = u i u ∗ i . Then { E , E , E , E , E } is a COSI set and E i = circ (1 , ω i , ω i , ω i , ω i ) where ω = e i π isa primitive th root of . Now combine { E , E } and { E , E } to get the COSI set S = { E , E ′ , E ′ } where E ′ = E + E , E ′ = E + E . The elements in this COSI set are circulant matrices also but in additionhave real entries: E ′ = circ (1 , cos θ, cos 2 θ, cos 3 θ, cos 4 θ ) , E ′ = circ (1 , cos 2 θ, cos 4 θ, cos θ, cos 3 θ ) where θ = π . It is noted that cos 4 θ = cos θ, cos 3 θ = cos 2 θ – which could be deduced from the fact that { E ′ , E ′ } are symmetric!Then S is used to design unitary and paraunitary matrices with real coefficients as for example E E ′ E ′ E ′ E ′ E E ′ E E ′ . The summation is not unique but a unique expression can be obtained by expressing the idempotent as the sum of rank 1idempotents with increasing initial zeros. xample 2.5 Take the columns { u , u , . . . , u } of the normalised Fourier × matrix and form E i = u i u ∗ i . Combine { E , E } and { E , E } to obtain the real COSI set S = { E , E ′ , E , E ′ } where E ′ = E + E , E ′ = E + E . Now a primitive th root of is ω = e i π = cos π + i sin π and cos π = , cos π = − . Hence E = circ (1 , , , , , , E = circ (1 , − , , − , , − , E ′ = circ (2 , , − , − , − , , E ′ = circ (2 , − , − , , − , − . S is used to form unitary and paraunitary matrices with real coefficients. The same trick can be applied to the normalised Fourier n × n matrix to obtain COSI sets with realcoefficients.Paraunitary and unitary matrices of size kn × kn are designed from a COSI set { E , E , . . . , E k } of k elements of size n × n by Propositions 2.1, 2.2. The following constructs paraunitary and unitary matricesof size n × n from a COSI set of size n × n . Proposition 2.3 [7]. Let { E , E , . . . , E k } be a COSI set.(i) Define U ( z ) = k X j =1 ± E j z t j . Then U ( z ) U ∗ ( z − ) = I .(ii) Let z = ( z , z , . . . , z k ) and U ( z ) = k X j =1 E j z j . Then U ( z ) U ∗ ( z − ) = I .(iii) Define U ( z ) = k X j =1 e iθ j E j z t j . Then U ( z ) U ∗ ( z − ) = I . When the z is replaced by an element of modulus 1 in (i) of Proposition 2.3, a unitary matrix is obtained.Other versions of Proposition 2.3 may be formulated, for example letting some of the z j in (ii) be equaland/or letting some be powers of the others.Using COSI sets to design paraunitary matrices is developed further in section 2.6 and using the sets toconstruct types of Hadamard matrices is developed in section 2.7. In these sections, the COSI methods arecombined with the designs methods of section 2.2. U is a symmetric unitary matrix if and only if U = ( I − E ) where E is a (symmetric) idempotent, see [13].This gives a method for constructing a unitary symmetric matrix from any idempotent.Let U = ( I − E ) where E is a symmetric idempotent. { E, I − E } is a COSI set. Note ( I − E ) E = − E, ( I − E )( I − E ) = I − E and thus ( I − E ) has eigenvalue − E and has eigenvalue 1 occurring to multiplicity equal to rank( I − E ).The requirement that U be of a particular type of symmetric matrix can be more difficult. Now H is asymmetric Hadamard matrix if and only if U = √ n H is a symmetric unitary matrix if and only if this U has a form ( I − E ) for a symmetric idempotent E . Thus a search for symmetric Hadamard matrices couldbegin with a search for such idempotents.Suppose U is any unitary matrix. Then its columns { u , u , . . . , u n } give rise to the COSI set { E , E , . . . , E n } with E i = u i u ∗ i . Some of the E i may be combined to form different COSI sets: P ki =1 E j i is also a symmetricidempotent, with J = { j , j , . . . , j k } ⊂ { , , . . . , n } , and this idempotent is orthogonal to each { E j | j J } or any idempotent formed in this way from { E j | j J } . Example 2.6
Let E = , F = ω ω ω ωω ω where ω is a primitive rd root of unity. hen { E, F } are idempotents and U = ( I − E ) , V = ( I − F ) are unitary matrices. Note K = √ U satisfies KK ∗ = K = 3 I but is not a Hadamard matrix. Also U V = V U as E, F are orthogonal.
Infinite series of symmetric unitary matrices may be obtained as illustrated in the following example.
Example 2.7
Let { E , E } be a COSI set. Then U = (cid:18) E E E E (cid:19) is a symmetric unitary matrix. Thus F = ( I − U ) , F = ( I + U ) is an orthogonal set of idempotents and so U = (cid:18) F F F F (cid:19) is a symmetricunitary matrix. Then { ( I − U ) , ( I + U ) } is a COSI set with which to form a symmetric unitary matrix.This process may be continued to produce an infinite series of symmetric unitary matrices.Initial choices for { E , E } include { E = (cid:18) (cid:19) , E = (cid:18) − − (cid:19) } and { E = (cid:18) i − i (cid:19) , E = (cid:18) − ii (cid:19) } . The { E , E } can be of any size and not just × matrices and any COSI set may be usedinitially. Example 2.8
Let U = √ (cid:18) −
11 1 (cid:19) . Define E = u u ∗ = (cid:18) (cid:19) , E = u u ∗ = (cid:18) − − (cid:19) where { u , u } are the columns of U . Then U ( z ) = E z i + E z j is a paraunitary matrix. U ( z ) has real entries andis symmetric in that U ( z ) ∗ = U ∗ ( z − ) = U ( z − ) . Multiplying any of the form U ( z ) using the same COSIset gives another of this form. However different COSI sets may be used to form paraunitary of the form U ( z ) and these may be combined to give different types of paraunitary matrices. Example 2.9
This gives an example of the design of a filter bank from COSI sets. A unitary real × matrix is of the form (cid:18) cos θ sin θ − sin θ cos θ (cid:19) . The above matrix U is of this form where θ = − π . Define E = (cid:18) cos θ − cos θ sin θ − sin θ cos θ sin θ (cid:19) , E = (cid:18) sin θ cos θ sin θ sin θ cos θ cos θ (cid:19) . Then { E , E } is a COSI set and U ( z ) = E z i + E z j is a paraunitary matrix. Different U ( z ) are obtained by taking different values of θ and thesecan then be used to design other paraunitary matrices of different forms. Paraunitary matrices of the type A + A z + . . . + A n − z n − are obtained with real coefficients. From this a 2-channel filter bank with n taps may be constructed. The primitive central idempotents, see [2], of the group ring C G form a complete orthogonal set of idempotentsand these can be realised as a COSI set in C n × n where n is the order of the group G . Interesting unitary andparaunitary matrices may be formed from the group ring C G of a finite group. The unitary and paraunitarymatrices formed have rational coefficients when G = S n , the symmetric group on n letters, and have realcoefficients when G = D n the dihedral group of order 2 n . Central primitive idempotents may also becombined to give a COSI with real entries as the idempotents occur in types of conjugate pairs. Someexamples may be found in [7]. The group ring aspects need to be investigated further; some ideas for thispaper occurred while looking at COSI sets in group rings. The following constructions were initiated by Dit¸˘a, [10], where it is essentially set up in order to buildHadamard matrices from lower order Hadamard matrices. It has been rediscovered in various forms a number6f times including by us. The original definition involved square matrices only and it is here generalisedslightly to work for non-square matrices and with two ‘sides’.
Definition 2.1 (Dit¸˘a [10]) Let { A , A , . . . , A k } be m × n matrices and let U = ( u ij ) be a k × k matrix.Define the left matrix tangle product of { A , A , . . . , A k } relative to U to be the mk × nk matrix A u A u . . . A k u k A u A u . . . A k u k ... ... . . . ... A u k A u k . . . A k u kk and the right matrix tangle product of { A , A , . . . , A k } relative to U to be the mk × nk matrix A u A u . . . A u k A u A u . . . A u k ... ... . . . ... A k u k A k u k . . . A k u kk .The notation ( U ; A , A , . . . , A k ) is used for the left matrix tangle product and ( A , A , . . . , A k ; U ) is usedfor the right matrix tangle product. From the context it will often be clear which (left or right) matrix tangleproduct is being used and in this case the term matrix tangle product is utilised. The Dit¸˘a construction as in [10, 9, 4] is given as a left matrix tangle product with square matrices. Theright tangle product is not the same as (is not equal to) the left tangle product but ( A , A , . . . , A k ; U ) =( U T ; A , A , . . . , A k T ) T for square matrices. It is convenient here for applications to have both left andright constructions and also for constructions when the matrices are not square as in Definition 2.2 below.A generalised version of this construction has also been used but this is not needed here. The presentconstructions are used with a view to producing non-separable constructions in particular.Definition 2.2 can be generalises as follows to the case where U is not square but has size either k × n or n × k where k is the number of matrices to be entangled; this requires the left or right matrix tangle productdefinitions. Definition 2.2 (i) Let { A , A , . . . , A k } be m × n matrices and let U = ( u ij ) be a t × k matrix. Define theleft matrix tangle product of { A , A , . . . , A k } relative to U to be the tm × nk matrix A u A u . . . A k u k A u A u . . . A k u k ... ... . . . ... A u t A u t . . . A k u tk (ii) Let { A , A , . . . , A k } be m × n matrices and let U = ( u ij ) be a k × t matrix. Define the right matrixtangle product of { A , A , . . . , A k } relative to U to be the mk × nk matrix A u A u . . . A u t A u A u . . . A u t ... ... . . . ... A k u k A k u k . . . A k u kt The notation ( U ; A , A , . . . , A k ) is used for the left matrix tangle product and ( A , A , . . . , A k ; U ) is usedfor the right matrix tangle product. From the context it may be clear which (left or right) tangle product isbeing used and in this case the term matrix tangle product is used. The matrix tangle product depends on the order of the A i and different tangle products are obtainedfrom different permutations of the A i - ‘different permutations’ should take into account that some of the7 i may be the same. This can be particularly useful in designing series of different matrices with desiredproperties which are non-separable (entangled).If all the A i = A are the same then the matrix tangle product is the matrix tensor product U ⊗ A . Thedirect sum of matrices is also a very special matrix tangle product as (cid:18) A B (cid:19) = ( I ; A, B ).Say U is the shuffler matrix and say { A , A , . . . , A k } are the tangle matrices of the matrix tangle product( U ; A , A , . . . , A k ) or of ( A , A , . . . , A k ; U ) depending on which, left or right, matrix tangle product is underconsideration. Suppose now an m × n matrix U is to be a shuffler matrix of a matrix tangle product. Theneither m or n matrices are required for the tangles but they need not all be different. If they are all the sameand equal to A then the tensor product U ⊗ A is obtained which is an mt × nq matrix when A is t × q . Ifless than n or m different matrices are to be used as tangles then these are repeated until m or n matricesare obtained as appropriate.The matrix tangle product may be square even though neither the tangles nor the shuffler are square.For example if { A, B } are 2 × U is a 3 × U ; A, B ) is a 6 × { A , A , . . . , A k } are k × t matrices and U is t × k then ( U ; A , A , . . . , A k ) is a kt × kt matrix; if { A , A , . . . , A k } are t × k matrices and U is k × t then ( A , A , . . . , A k ; U ) is a kt × kt matrix.The matrix tangle product is not a matrix tensor product unless each A i = α i A some α i and some fixed A . In this situation ( U ; α A, α A, . . . , α k A ) = ( U ′ ; A, A, . . . , A ) = U ′ ⊗ A where U ′ is obtained from U bymultiplying rows or columns of U by appropriate α i .The matrix tangle product has some linearity: • α ( U ; A , A , . . . , A k ) = ( U ; αA , αA , . . . , αA k ) = ( αU ; A , A , . . . , A k ). • ( U + V ; A , A , . . . , A k ) = ( U ; A , A , . . . , A k ) + ( V ; A , A , . . . , A k ). • ( U ; A , A , . . . , A k ) + ( U ; B , B , . . . , B k ) = ( U ; A + B , A + B , . . . , A k + B k ).Similar results hold for the right matrix tangle product.Note however for example that ( U ; A + A ′ , A ) is not in general the same as ( U ; A , A ) + ( U ; A ′ , A ).The determinant value of a matrix tensor product of square matrices in terms of the constituents isinteresting. It is ‘preserved’ or rather can be obtained in terms of the determinants of the tangles andshuffler, see subsection 2.3 below, Proposition 2.3. However the spectrum does not have a relationship withthe spectrums of the constituents, as happens for a matrix tensor product, due to ‘entangling’. Let T = ( U ; A , A , . . . , A k ). It is of interest to know the value of det T = | T | when the A i and U are squarematrices. It’s given in terms of the determinants of the constituents as follows. Proposition 2.4
Let T = ( U ; A , A , . . . , A k ) where U is a k × k matrix and the A i are n × n matrices.Then | T | = | A || A | ... | A k || U | n . Proof:
This can be shown using results on determinants of block matrices as for example in [1]. Alternativelya direct proof may be given by applying the techniques used in such results on block matrices. If A = 0or if all α i = 0 the result is clear. An inductive argument is used by applying block operations on T toreduce the first column of blocks to the form α A where U = ( α ij ); these block operations do not alter Matrix tensor product is often called
Kronecker product . See however [5] for discussion on this name. α = 0 we can assume α i = 0 for some α i and proceed similarly.Then | T | = det( α A ) × | B | where B is a similar matrix but of one block size smaller and induction maybe applied. (cid:3) A similar result holds for the right matrix tangle product.This property is particularly useful in applications, see for example Section 3.Proposition 2.4 generalises the determinant value of a matrix tensor product – if all the A i are the same, A i = A , then | T | = | A | k | U | n and T = U ⊗ A .For example let { A, B } be n × n matrices and let U be of size 2 ×
2. Then T = ( U ; A, B ) has | T | = | A || B || U | n .Finding the eigenvalues of a matrix tangle product is difficult and no formula in terms of the eigenvaluesof the constituents exists. The eigenvalues of a matrix tangle product are ‘entangled’. Which properties of the shuffler and tangles of a matrix tangle product are preserved? Let P be a propertyof a matrix, such as for example being unitary or invertible. Say the matrix M ∈ P if and only if M hasthis property P . If for any G = ( U ; A , A , . . . , A k ) with A i ∈ P for i = 1 , , . . . , k and U ∈ P , implies that G ∈ P then say the matrix tangle product preserves P . • ‘Unitariness’ is preserved. • ‘Invertibility’ is preserved. • ‘Paraunitariness’ is preserved. • ‘Normality’ is not preserved. • ‘Being Symmetric’ is not preserved. • ‘Being Hadamard’ is preserved.The preserved properties are stated as Propositions in the following subsections 2.5, 2.6 and 2.7, andthese subsections derive applications and constructions as relevant to the subsections. Proposition 2.5
Let { A , A , . . . , A k } be m × m unitary matrices and let U = ( u ij ) be a unitary k × k matrix. Then A u A u . . . A k u k A u A u . . . A k u k ... ... ... ... A u k A u k . . . A k u kk and A u A u . . . A u k A u A u . . . A u k ... ... ... ... A k u k A k u k . . . A k u kk are unitary mk × mk matrices. The matrix tangle products of unitary matrices are unitary matrices. Section 2.1 also constructs unitarymatrices from COSI, complete orthogonal symmetric idempotent, sets. This greatly expands the pools ofunitary matrices available for various purposes. Non-separable, entangled, matrices are often required. Amatrix is unitary if and only if its rows or columns form an orthonormal basis and thus new orthonormalbases are constructed when a new unitary matrix is constructed.9 .5.1 Pauli unitary matrices as builders
Applying the process to the Pauli matrices σ x = ( ) , σ y = (cid:0) i − i (cid:1) , σ z = (cid:0) − (cid:1) gives interesting entangledunitary matrices. The following six 4 × σ z ; σ x , σ y ) , ( σ z ; σ y , σ x ) , ( σ y ; σ x , σ z ) , ( σ y ; σ z , σ x ) , ( σ x ; σ z , σ y ) , ( σ x ; σ y , σ z )Other 4 × { σ x , σ y , σ y } ; some are tensor products such as( σ x , σ x ; σ z ) and ones are like ( σ x , σ y ; σ x ) where a matrix appears as both a tangle and the shuffler.Taking two of these 4 × { σ x , σ y , σ z } as a shuffler producesan 8 × n × n unitary entangled matrices from the Pauli matrices.The significance of these needs to be explored. Start with the following real 2 × (cid:0) − (cid:1) , (cid:0) −
11 1 (cid:1) , (cid:0) − (cid:1) , (cid:0) − (cid:1) from which to build new matrices.Make these unitary by dividing by √ (cid:0) cos θ sin θ − sin θ cos θ (cid:1) are often used in practice. Different θ maybe used from which real 2 n × n real orthogonal matrices are built.Now A i = e iθ i are 1 × U be a k × k unitary matrix. Then ( U ; A , A , . . . , A k ) , ( A , A , . . . , A k ; U ) are also unitary k × k matrices. Example 2.10 • Let U = √ (cid:18) − (cid:19) and let A = (1) , B = ( i ) . Then U, A, B are unitary matri-ces. Now ( A, B ; U ) is a unitary matrix G = √ (cid:18) i − i (cid:19) . Then { U, G, I } constitute three matricesconsisting of mutual unbiased bases for C . • Let U = √ ω ω ω ω . Let A = (1) , B = ( ω ) , C = ( ω ) and form U = ( A, B, C ; U ) . Let A = (1) , B = ( ω ) , C = ( ω ) and form U = ( A, B, C ; U ) . Then { U, U , U , I } are 4 matricesconsisting of mutually unbiased bases for C . Paraunitary matrices are fundamental in signal processing and the concept of a paraunitary matrix playsan important role in the research area of multirate filterbanks and wavelets. In the polyphase domain, thesynthesis matrix of an orthogonal filter bank is a paraunitary matrix; a Filter Bank is orthogonal if itspolyphase matrix is paraunitary, see [3]. Thus designing an orthogonal filter bank is equivalent to designinga paraunitary matrix. The book [3], Chapters 4-6, makes the design of paraunitary matrices a primary aim.
Non-separable/entangled paraunitary matrices are often a requirement.The literature is huge and expanding rapidly; of particular note is [11] where further background and manyreferences therein may be found. Quotations from the literature: “Designing nonseparable multidimensionalorthogonal filter banks is a challenging task.”; “Multirate filter banks give the structure required to generateimportant cases of wavelets and the wavelet transform.”; “ In filter bank literature the terms orthogonality,paraunitary and lossless are often used interchangeably.” “Paraunitryness is a necessary and sufficientcondition for wavelet orthogonality.” “Designing an orthogonal filter bank is equivalent to designing aparaunitary matrix.” 10araunitary matrices are constructed using COSI sets by methods of Propositions 2.2 and 2.3, see Section2.1; paraunitary matrices which are symmetric may be built with this method.‘Being a paraunitary matrix’ is a property preserved by matrix tangle products.
Proposition 2.6
Let { A , A , . . . , A k } be m × m paraunitary matrices and let U = ( u ij ) be a paraunitary k × k matrix. Then A u A u . . . A k u k A u A u . . . A k u k ... ... ... ... A u k A u k . . . A k u kk and A u A u . . . A u k A u A u . . . A k u k ... ... ... ... A k u k A k u k . . . A k u kk are paraunitary mk × mk matrix in the union of the variables in { A , A , . . . , A k , U } . The constructions in Propositions 2.2, 2.3 and 2.6 may be combined. Building blocks for paraunitarymatrices are available; these are not tensor products. The shuffler itself may be a unitary matrix as mayany of the tangles. Examples are given in [7] where a more restricted restricted tangle definition is given.Although the systems here give building blocks for multidimensional paraunitary matrices, it is not claimedthat every multidimensional paraunitary matrix is built in this way. The renowned building blocks for 1Dparaunitary matrices over C due to Belevitch and Vaidyanathan as described in [16] are constructed from acomplete orthogonal idempotent set of two elements.Now A i = z i are 1 × P be a k × k paraunitary matrix. Then G =( P ; A , A , . . . , A k ) is a paraunitary k × k matrix in the union of the variables in P and { z , z , . . . , z k } .By replacing the variables by elements of modulus 1 in a paraunitary matrix, a unitary matrix is obtained.Thus constructing paraunitary matrices leads to the construction of unitary matrices. ↔ Unitary H is a Hadamard n × n matrix if its entries are elements of modulus 1 and HH ∗ = I n . A matrix of type H ( n, p ) is a Hadamard matrix in which each element is a p th root of 1. A H ( n,
2) matrix is a real Hadamardmatrix. It is known that the Dit¸˘a construction preserves Hadamard matrices, [10, 9, 4].
Proposition 2.7 [10] Let { A , A , . . . , A k } be m × m Hadamard matrices and let U = ( u ij ) be a Hadamard k × k matrix. Then(i) A u A u . . . A k u k A u A u . . . A k u k ... ... ... ... A u k A u k . . . A k u kk is a Hadamard km × km matrix. If the A i and U have entries whichare n th roots of then this matrix has entries which are n th roots of .(ii) A u A u . . . A u k A u A u . . . A u k ... ... ... ... A k u k A k u k . . . A k u kk is a Hadamard km × km matrix. If the A i and U have entrieswhich are n th roots of then this matrix has entries which are n th roots of . The Dit¸˘a product has been used in a number of papers to construct Hadamard matrices from lower orderHadamard matrices, see for example [10] itself, [9] and [4]. Hadamard matrices have been also constructedin section 2.1 by the COSI method.Now A i = e iθ i are 1 × H is a H ( n, p ) matrix if it has size n and entries whichare p th roots of 1. Let H be a k × k Hadamard matrix. Then G = ( H ; A , A , . . . , A k ) is a Hadamard matrix.11f H = H ( k, p ) and { A i = A i (1 , p ) } then G is a G ( k, p ) matrix. If H = H ( k, p ) and A i = A i (1 , n i ) then G is a G ( k, s ) matrix where s = lcm( p, n , n , . . . , n k ).Symmetric Hadamard matrices are Type II matrices; the definition and further information on Type IImatrices may be found in [9] and the many references therein. “Type II matrices were introduced explicitlyin the study of spin models. ” The following construction is similar to that formulated in for example [4]but is a useful way with which to look at the formulation of symmetric Hadamard matrices.
Construction 2.1
Let H be a Hadamard matrix of type H ( n, p ) . Let G be the corresponding unitary matrix,that is G = √ n H . The columns { u , u , . . . , u n } of G form an orthonormal basis for C n . Let E i = u i u ∗ i .Then { E , E , . . . , E n } is a COSI set, from which unitary n × n matrices may be formed as in section 2.1.In particular symmetric n × n matrices may be formed using the reverse circulant construction. Thesematrices have entries which are n times a p th root of and so multiplying any of these matrices by n givesa symmetric n × n Hadamard matrix which is a H ( n , p ) matrix. Starting from any
Hadamard H ( n, p ) matrix Hadamard H ( n , p ) matrices are produced some of which aresymmetric. The process may then be continued to produce H ( n k , p ), for k ≥
1, unitary and Hadamardmatrices. By taking the reverse circulant process at any stage of production the matrices produced aresymmetric – only at the final stage need the reverse circulant process be applied in order to get symmetricmatrices.It is also known that a symmetric 2 n × n Hadamard symmetric matrices may be constructed from an n × n symmetric matrices, see [4]. The construction 2.2 below is more natural and illustrative. Construction 2.2 (i) Let H be an n × n Hadamard symmetric matrix and U a × symmetric matrix.Then ( U ; A, A T ) , ( U ; A T , A ) , ( A, A T ; U ) , ( A T , A : U ) are symmetric Hadamard n × n matrices.(ii) Let H be an n × n Hadamard symmetric H ( n, p ) matrix and U a × symmetric U (2 , p ) matrix. Then ( U ; A, A T ) , ( U ; A T , A ) , ( A, A T ; U ) , ( A T , A : U ) are symmetric Hadamard n × n matrices of type G (2 n, p ) .More generally if H is of type H ( n, p ) and U is of type U (2 , q ) then ( U ; A, A T ) , ( U ; A T , A ) , ( A, A T ; U ) , ( A T , A : U ) are of type G (2 n, s ) where s = lcm ( q, p ) . The n × n Fourier unitary is a Hadamard H ( n, n ) matrix. Example 2.11
Let H = ω ω ω ω where ω is a primitive third root of . Then G = √ H is aunitary matrix. The columns of G are u = √ (1 , , T , u = √ (1 , ω, ω ) T , u = √ (1 , ω , ω ) T . Then { E = u u ∗ = , E = u u ∗ = ω ωω ω ω ω , E = u u ∗ = ω ω ω ωω ω } is a COSIset. Thus K = E E E E E E E E E is a symmetric unitary matrix and L = 3 K is a symmetric Hadamard L (9 , matrix. Example 2.12 : P = (cid:0) − (cid:1) , Q = ( ii ) . are Hadamard H (2 , matrices. Then A = √ P, B = √ B are unitary matrices. Infinite series of unitary and Hadamard matrices may be built as follows. Build { A, B } relative to unitary A and then build { A, B } relative to unitary B to obtain Build A = ( A, B ; A ) = (cid:18) i − i
11 1 − − i − − i (cid:19) , B = ( A, B ; B ) = (cid:18) i − − − ii i ii − i i (cid:19) . Other options for A , B are A = ( A ; A, B ) , B =( B ; A, B ) but also others such as swapping A, B around. These are × matrices and A , B are Hadamard H (4 , matrices. uild ( A , B ; A ) , ( A , B ; B ) to get unitary × matrices with entries ± , ± i and from these get H (8 , matrices. Build ( A , B ; A ) , ( A , B ; B ) to get H (16 , matrices. The process may be continued in manydifferent directions. A Hadamard n × n matrix is skew symmetric provided H + H T = 2 I n . The Dit¸˘a product may be usedto produce skew symmetric 2 n × n Hadamard matrices from a skew symmetric n × n Hadamard matrix.Skew symmetric Hadamard matrices are used for various designs and for constructing symmetric conferencematrices.
Construction 2.3
Let A be an n × n skew symmetric Hadamard matrix and let U be a × skew symmetricHadamard matrix. Then ( U ; A, A T ) , ( U ; A T , A ) , ( A, A T ; U ) , ( A T , A ; U ) are skew symmetric n × n matrices. The normal method, see for example [4], for producing a 2 n × n skew symmetric matrix from a n × n skewsymmetric matrix is a special case of Construction 2.3.More generally say H is an n × n skew Hadamard matrix, over C , provided H is a Hadamard matrix and H + H ∗ = 2 I n ; also assume the diagonal entries of H are 1. Construction 2.3 works for general Hadamardskew matrices where the transpose, T , is replaced by complex conjugate transpose, ∗ .The 2 × (cid:0) −
11 1 (cid:1) or (cid:0) − (cid:1) or similar. A skew Hadamard2 × C in addition are ones of the form (cid:0) α − α ∗ (cid:1) where | α | = 1.Suppose now H = e iα − e iα − e iα − e − iα − e − iα e iα e iα e iα e iα e iα − e − iα − e − iα is to be a Hadamard matrix; it already has theskew condition. Then looking at HH ∗ = 4 I n the following conditions are obtained: (i) − α + α = − α − α ;(ii) α − α = − α − α ; (iii) α + α = α − α . Solving this system of equations gives α = α + α , α = − α − α and α , α , α can have any value. This gives an infinite number of skew Hadamard (complex)matrices. New infinite sets can be formed using Construction 2.3.As an example require the { e iα j } to be n th roots of 1. Say for example α = πn , α = πn , α = πn andthen α = πn , α = − πn , α = − πn .This gives the following skew Hadamard matrix ω − ω − − ω − − ω − − ω − ω − ω ω ω − ω − ω − ω , where ω = e i πn is aprimitive n th root of 1. Taking ω to be a primitive third root of unity, ω = 1, gives the skew Hadamardmatrix ω − ω − − ω − ω ω ω − ω − ω . The entries are 6 th roots of unity, so this is a H (4 ,
6) matrix.A unitary matrix may be obtained directly from any Hadamard matrix: If H is a Hadamard n × n matrixthen U = √ n H is a unitary matrix. The entries of this corresponding unitary matrix have a special form.Infinite sequences of skew Hadamard real matrices may be obtained by starting out with a skew Hadamardmatrices A and with U = (cid:0) −
11 1 (cid:1) or U = (cid:0) − (cid:1) .Then form A which can be one of ( U ; A, A T ) , ( U ; A T , A ) , ( A, A T ; U ) , ( A T , A ; U ).Replace A by A to form one of ( U ; A , A ) , ( U ; A , A ) , ( A , A ; U ) , ( A , A ; U ) and so on.Let A be a normalised n × n Fourier matrix and B a matrix obtained from A by interchanging rows (orcolumns). Then both A, B are unitary matrices. Let C be any 2 × A, B ; C ) and13 B, A ; C ) are unitary 2 n × n matrices. Let A be a Hadamard matrix and B any permutation of the rows ofcolumns of A . Let C be any 2 × A, B ; C ) and ( B, A ; C ) are Hadamard matrices.If A is of type H ( n, q ) and C is of type H (2 , q ) then type of ( A, B ; C ) , ( B, A ; C ) have a determined type.As an explicit example consider the following: Let A = √ ω ω ω ω , B = √ ω ω ω ω , C = √ (cid:18) − (cid:19) , where ω is a primitive 3 rd root of unity.Then ( A, B ; C ) , ( B, A ; C ) are 6 × α = √ times 6 th roots ofunity and so α ( A, B ; C ) , α ( B, A ; C ) are Hadamard matrices with entries which are 6 th roots of unity.This can also be played out for the discrete cosine and sine transforms. Let A, B be discrete transformsand C any 2 × { ( A, B ; C ) , ( B, A ; C ) } are multidimensional transforms which are notmatrix tensor products.Hadamard matrices have been designed from matrix tensor products – if A, B are Hadamard matrices so is A ⊗ B . Many formulations of Hadamard constructions are equivalent to matrix tensor product constructions.Thus tangle product generalises the matrix tensor product method for constructing Hadamard matrices;the matrix tensor product method includes Sylvester’s method. Sylvester’s method for producing Walshmatrices starts out with U = (cid:18) − (cid:19) and goes to (cid:18) A AA − A (cid:19) where A has already been constructed; thisis A ⊗ U . A similar series may be obtained by starting out with for example beginning with the same ordifferent initial U and then producing ( A, B ; U ) from previously produced A, B . Indeed the U could changeat any stage. The Walsh-Hadamard transfer has uses in many areas and is formed using a matrix tensorproduct starting out with (cid:18) − (cid:19) . Many variations on this may be obtained using matrix tangle products;for instance the related matrices (cid:18) − (cid:19) , (cid:18) −
11 1 (cid:19) , (cid:18) − (cid:19) , (cid:18) − (cid:19) could be used and entangled.Hadamard matrices can also be designed from paraunitary matrices which themselves have been designedby orthogonal symmetric complete sets of idempotents, see section 2.1. Section 2.1 initiates COSI constructions and Section 2.2 deals with Dit¸˘a type constructions. The two may becombined to derive further builders. The COSI construction can be used to construct unitary, paraunitaryor Hadamard matrices and these may then be used to construct matrix types using the Dit¸˘a construction.On the other hand suppose a unitary matrix is constructed by either method. Then the columns of thematrix may be used to construct COSI sets from which further unitary, paraunitary or other matrix typescan be constructed by the COSI method of section 2.1.
Example 2.13
Let U = √ (cid:0) −
11 1 (cid:1) , A = √ (cid:0) − (cid:1) , B = √ (cid:0) i − i (cid:1) . Then ( U ; A, B ) = (cid:18) − − − − i i − i − i (cid:19) . × ( U ; A, B ) is a Hadamard H (4 , matrix.Let F = u u ∗ , F = u u ∗ u , F = u u ∗ , F = u u ∗ where { u , u , u , u } are the columns of ( U ; A, B ) .Then F α F α F α F α F α F α F α F α F α F α F α F α F α F α F α F α , for variables α i , is a paraunitary matrix which is a unitarymatrix when the variables are given values of modulus . Also F α + F α + F α + F α is a paraunitarymatrix when the variables are given values of modulus .The process may be continued and infinite sequences obtained. .8.1 Infinite sequences Let P be a property such that the matrix tangle property preserves P . Infinite series of non-separablematrices with property P may be obtained in many ways and in many directions. Here we give somegeneral methods. Example 2.13 above gives the flavour. The methods lead easily to strong encryptiontechniques including public key systems. Error correcting codes may also be developed and both encryptionand error-correcting may be included in the one system.Consider constructing infinite sequences of non-separable matrices with property P using initially just twomatrices with property P . Let A , A are 2 × P which is preserved by matrix tangleproduct. Form the 4 × A ; A , A ) = A , ( A ; A , A ) = A , ( A ; A , A ) = A , ( A ; A , A ) = A which then have property P . Each of the 12 pairs { A i , A j | i = j } may be tangleswith shuffler A or A giving 24 new entangled matrix tangle products of size 8 × P . Choose2 different elements of these 24 and form tangle products with either A or A to get 16 ×
16. This can becontinued indefinitely. At each stage, matrices with property P are obtained.Infinite series with real entries may be obtained. The initial matrices could be real orthogonal as forexample A = √ (cid:18) − (cid:19) , A = √ (cid:18) − (cid:19) or more generally of the form (cid:18) cos θ sin θ − sin θ cos θ (cid:19) for differing θ . Let S = { A , A , . . . , A k } be a set of size t × t matrices with property P . Construct ( U ; A i , A i , . . . , A i n )or ( A i , A i , . . . , A i n ; U ) with i j ∈ { , , . . . , k } and U some n × n matrix with property P . For example P could be the property of being unitary and U could be the n × n unitary Fourier matrix. To be non-separableit is necessary that the i j not all be equal. This constructs nt × nt matrices with property P ; the U can varyalso. Infinite series are obtained by varying n . Many such different infinite sequences may be constructed. In section 2 construction methods were laid out for various types of matrices and applications to unitary,paraunitary and special types of these were given. Here we give separately an application to the design ofconstellations of matrices. The design problem for unitary space time constellations in set out as follows as in[12] and [6]: “Let M be the number of transmitter antennas and R the desired transmission rate. Constructa set V of L = 2 RM unitary M × M matrices such that for any two distinct elements A, B in V , the quantity | det( A − B ) | is as large as possible. Any set V such that | det( A − B ) | > A, B is said tohave full diversity .”The number of transmitter antennas is the size M of the matrices.The set V is known as a constellation and the quality of the constellation is measured by ζ V = 12 min V l ,V m ∈V ,V l = V m | det( V l − V m ) | M Methods for constructing constellations while determining their quality using orthogonal symmetric idem-potent sets was initiated in [8]. These can now be expanded and further constellations obtained using theconstructions in Section 2.The survey article [15] proposes division algebras for this area and some comparisons can be made.Let { A , A , . . . , A k } be a constellation of m × m matrices with quality ζ and let U be a unitary matrix.Then1. { ( U ; A i , A i , . . . , A i k ) | ( i , i , . . . , i k ) is a derangement of(1 , , . . . , k ) } is a constellation of mk × mk matrices of quality ζ . A derangement is a permutation such that no element appears in its originalposition. 15. Let { U i | i = 1 , , . . . s } be a constellation of quality ζ of k × k matrices and { A , A , . . . , A k } any k unitary t × t matrices. Then { ( U i ; A , A , . . . , A k ) | i = 1 , , . . . , s } is a constellation of kt × kt matriceswith quality also ζ .Unitary matrices and paraunitary matrices are constructed according to Proposition 2.2 using a completeorthogonal idempotent (COSI) set { E , E , . . . , E k } and forming G = E i α E i α . . . E i k α k E i α E i α . . . E i k α k ... ... ... ... E i k α k E i k α k . . . E i kk α kk where { E , E , . . . , E k } appear once in each row and column.Let G = (cid:18) E α E α E α E α (cid:19) where E , E is an COSI 2 × α i are elements in C . Thendet G = α α . Let now α i be n th roots of unity and then { (cid:18) E α E α E α E α (cid:19) } is a constellation which hasfull diversity when an n th root of 1 appears just once in each block column. Let A = (cid:18) E α E α E α E α (cid:19) , B = (cid:18) E β E β E β E β (cid:19) . Then | det( A − B ) | = | ( α − β ) ( α − β ) | = | ( α − β ) | | ( α − β ) | .The following is well-known and is easily verified. Lemma 3.1
Let z = cos θ + i sin θ . Then | − z | = 2 | sin θ | Corollary 3.1
Let α = ω i , β = ω j with i = j and ω = e iπn is a primitive n th root of unity. Then | α − β | = 2 | sin θ | where θ = π ( j − i ) n . Now from Corollary 3.1, | det( A − B ) | ≥ | sin θ | where θ = πn . Thus the quality of the constellation is (2 ( | sin θ ) | ) = | sin θ | .The number that can be in each constellation when n th roots of unity are used is n . For n = 4, θ = π and the quality ≈ . .. ; the rate is . For n = 8, θ = π and the quality is ≈ . ... ; the rate is .For n = 16, θ = π and the quality is ≈ . ... ; the rate is 1.Higher order constellations may also be deduced and quality determined as follows.Let G = E α E α . . . E n α n E n α E α . . . E n − α n ... ... ... ... E n α E n − α . . . E α n where { E , E , . . . , E n } is a COSI set and the α i areelements in C . Then it transpires that | det( G ) | = | α α . . . α n | n , where n is the size of the matrix E j . Theset E α E α . . . E n α n E n α E α . . . E n − α n ... ... ... ... E n α E n − α . . . E α n with the | α i | = 1 is then a constellation (of unitary matrices).Let the α j be n th of unity such that no α j appears in more than one block column. Then the quality of thisconstellation is | sin θ | where θ = πn . There can be up to n elements in the constellation. References [1] Philip Powell, ‘Calculating Determinants of Block Matrices’, arXiv:1112.4379162] C. P. Milies and S. K. Sehgal,
An introduction to Group Rings , Klumar, 2002.[3] Gilbert Strang and Truong Nguyen,
Wavelets and Filter Banks , Wesley-Cambridge Press, 1997.[4] C. J. Colbourn & J. H. Dinitz (Eds.),
Handbook of Combinatorial Designs , Discrete Mathematics andits applications, Chapman and Hall, London/New York, 2007.[5] H. Henderson, F. Pukelsheim, S. R. Searle, “On the history of the kronecker product”, Linear andMultilinear Algebra, Vol. 14,No.2, 113-120, 1983.[6] B. Hochwald, W. Sweldens, “Differential unitary space time modulation”, IEEE Trans. Comm., 48,(2000), 2041-2052.[7] Ted Hurley and Barry Hurley, “Paraunitary matrices and group rings”, Intl. J. Group Theory, Vol. 3,no.1, pp 31-56, 2015.[8] Ted Hurley, “Full diversity sets of unitary matrices from orthogonal sets of idempotents”,arXiv:1612.02202[9] R. Hosoya and H. Suzuki, “Type II Matrices and Their Bose-Mesner Algebras”, Journal of AlgebraicCombinatorics, 17, 19–37, 2003.[10] P. Dit¸˘a “Some results on the parametrization of complex Hadamard matrices”, J. Phys. A, 37 no.20,5355-5374, 2004.[11] Jianping Zhou, Minh N. Do, and Jelena Kovaˆcevi´c, “Special Paraunitary Matrices, Cayley Transformand Multidimensional Orthogonal Filter Banks”, IEEE Trans. on Image Processing, 15, no. 2, 511-519,2006.[12] A. Shokrollahi, B. Hassibi, B.M. Hochwald, W. Sweldens, “Representation theory for high-rate multiple-antenna code design”, IEEE Trans. on Inform. Theory, 47, no.6, (2001), 2335-2367.[13] Ted Hurley, “Unique builders for classes of matrices”, arXiv:1904.11250.[14] M. A. Nielsen, I. Chuang,
Quantum Computation and Quantum Information , Cambridge UniversityPress, Cambridge UK, 2010.[15] B.A. Sethuraman, “Division Algebras and Wireless Communication”, Notices of the AMS, 57, no. 11(2010), 1432-1439.[16] P. P. Vaidyanathan,