# XDistributedPolynomialFunctions(VAR, F)¶

Using the Hankel matrix of a noncommutative polynomial one can determine the rank (Matrices de Hankel by Fliess). The extended Ho-Algorithmus (Fornasini+Marchesini) is used to create a minimal linear representation. Another approach is implemented in NonCommutativeRationalFunctions.

display: Record(rows: List FreeMonoid VAR, cols: List FreeMonoid VAR, H: Matrix F) -> OutputForm

display(sys) prints the Hankel matrix in an extended form with row-indices as column 0 and column-indices as row 0.

findNonZeroEntry: (Matrix F, NonNegativeInteger, NonNegativeInteger) -> List NonNegativeInteger

findNonZeroEntry(A,i,j) returns the indices [k,l] with i<=k<=m and j<=l<=n such that A(k,l) is non-zero and [0,0] otherwise. A is of size m x n.

hankelIndices: XDistributedPolynomial(VAR, F) -> List FreeMonoid VAR

hankelIndices(p) returns a list of all left and right factors of the monomials of a given multivariate noncommutative polynomial. Factorization: w = u*v

hankelMatrix: (XDistributedPolynomial(VAR, F), VAR) -> Matrix F

hankelMatrix(p,x) returns a matrix with the entries of the coefficients of p at (u,v) where the monomials factorizes through w = u*x*v. This matrix is indexed by all words of the Hankel matrix.

hankelMatrix: XDistributedPolynomial(VAR, F) -> Matrix F

hankelMatrix(p) returns the Hankel matrix H(p) of given polynomial p, i.e. the entries at (u,v) are the coefficients of the monomials w = u*v. Rows and columns are indexed by words.

hankelSystem: XDistributedPolynomial(VAR, F) -> Record(rows: List FreeMonoid VAR, cols: List FreeMonoid VAR, H: Matrix F)

hankelSystem(p) creates a Hankel-matrix for the polynomial p with respect to the basis of all factors in p. Row- and column-indices can be different.

minimalMatrix: (Matrix F, NonNegativeInteger) -> Matrix F

minimalMatrix(A,r) returns the minimal (upper-left) submatrix of A such that the rank corresponds to the given.

minimalRepresentation: XDistributedPolynomial(VAR, F) -> Record(alpha: Matrix F, mu: List Matrix F, var: List VAR, beta: Matrix F)

minimalRepresentation(p) returns a minimal representation using the generalized Ho algorithm [Fornasini, 1978] p = sum_w alpha*mu(w)*beta*w [Theorem 3.3, Salomaa–Soittola 1978]

transformationMatrices: Matrix F -> List Matrix F

transformationMatrices(A) returns a list of two matrices P and Q such that P*A*Q = J where the upper diagonal block of J is I_k for k = rank(A).