The convert(p, MatrixPolynomialObject, x) function converts the (matrix or scalar) polynomial p into a standard representation, a Record. This allows systematic (conventional) access to the polynomial properties, such as Degree, in a manner independent of the polynomial basis.  The bases understood by MatrixPolynomialObject include:

and most others understood by the OrthogonalSeries package.  This routine is used internally by LinearAlgebra[CompanionMatrix].

If the input polynomial p contains more than one basis, then this (heuristic) conversion will fail.

The type(expr, MatrixPolynomialObject) function checks whether expr is a Record of the type returned by convert(...,MatrixPolynomialObject).

A MatrixPolynomialObject record has the following fields:

Basis - the name of the basis used; either PowerBasis or any of the supported basis names listed above.

BasisParameters - a list of the parameters of the particular basis; e.g. for LagrangeBasis or NewtonBasis these are the nodes; for BernsteinBasis these are the degree n and the left and right ends a and b of the interval.

Coefficient - a procedure to return a specific coefficient matrix. It takes as argument a nonnegative integer less or equal to Degree and returns a Matrix.

Degree - a nonnegative integer; the degree of the polynomial (in the LagrangeBasis or BernsteinBasis case, an upper bound on the degree).

Dimension - a positive integer; the matrix dimension $s$ of the matrix polynomial ($s=1$ if the original polynomial is a scalar polynomial).

IsMonic - a procedure without arguments returning true or false, depending on whether the polynomial is known to be monic (not relevant for Lagrange or Bernstein bases).

OutputOptions - a list of output options for the coefficient Matrices (see MatrixOptions).

Value - a procedure to evaluate the polynomial at any point. It takes as an argument the point (an algebraic expression) and returns a Matrix.

Variable - a name; the original variable used to define the polynomial (which may be unspecified in the LagrangeBasis case).

$p≔\mathrm{add}\left(\frac{k+1}{k+2}\mathrm{ChebyshevT}\left(k\,x\right)\,k=0..5\right)$

$p≔\frac{\mathrm{ChebyshevT}\left(0\,x\right)}{2}+\frac{2\mathrm{ChebyshevT}\left(1\,x\right)}{3}+\frac{3\mathrm{ChebyshevT}\left(2\,x\right)}{4}+\frac{4\mathrm{ChebyshevT}\left(3\,x\right)}{5}+\frac{5\mathrm{ChebyshevT}\left(4\,x\right)}{6}+\frac{6\mathrm{ChebyshevT}\left(5\,x\right)}{7}$

$P≔\mathrm{convert}\left(p\,\mathrm{MatrixPolynomialObject}\,x\right)$

$P≔\mathrm{Record}\left(\mathrm{Value}={\mathrm{Default}}_{\mathrm{value}}\,\mathrm{Variable}=x\,\mathrm{Degree}=5\,\mathrm{Coefficient}=\mathrm{coe}\,\mathrm{Dimension}=\left[1\,1\right]\,\mathrm{Basis}=\mathrm{ChebyshevT}\,\mathrm{BasisParameters}=\left[\right]\,\mathrm{IsMonic}=\mathrm{mon}\,\mathrm{OutputOptions}=\left[\mathrm{shape}=\left[\right]\,\mathrm{storage}=\mathrm{rectangular}\,\mathrm{order}=\mathrm{Fortran\_order}\,\mathrm{fill}=0\,\mathrm{attributes}=\left[\right]\right]\right)$

$P:-\mathrm{Degree}$

$5$

$P:-\mathrm{Dimension}$

$\left[1\,1\right]$

$P:-\mathrm{Value}\left(0.3\right)$

$\left[\begin{array}{c}0.5949161905\end{array}\right]$

$\mathrm{nodes}≔\left[-1\,-\frac{1}{3}\,\frac{1}{3}\,1\right]$

$\mathrm{nodes}≔\left[\mathrm{-1}\,-\frac{1}{3}\,\frac{1}{3}\,1\right]$

$\mathrm{values}≔\left[-1\,1\,-1\,1\right]$

$\mathrm{values}≔\left[\mathrm{-1}\,1\,\mathrm{-1}\,1\right]$

$P≔\mathrm{convert}\left(\mathrm{values}\,\mathrm{MatrixPolynomialObject}\,\mathrm{nodes}\right)$

$P≔\mathrm{Record}\left(\mathrm{Value}={\mathrm{Default}}_{\mathrm{value}}\,\mathrm{Variable}=\mathrm{\_X1}\,\mathrm{Degree}=3\,\mathrm{Coefficient}=\mathrm{coe}\,\mathrm{Dimension}=\left[1\,1\right]\,\mathrm{Basis}=\mathrm{LagrangeBasis}\,\mathrm{BasisParameters}=\left[\left[\mathrm{-1}\,-\frac{1}{3}\,\frac{1}{3}\,1\right]\right]\,\mathrm{IsMonic}=\mathrm{mon}\,\mathrm{OutputOptions}=\left[\mathrm{shape}=\left[\right]\,\mathrm{storage}=\mathrm{rectangular}\,\mathrm{order}=\mathrm{Fortran\_order}\,\mathrm{fill}=0\,\mathrm{attributes}=\left[\right]\right]\right)$

$P:-\mathrm{Degree}$

$3$

$P:-\mathrm{Value}\left(0.3\right)$

$\left[\begin{array}{c}\mathrm{-0.9284999999}\end{array}\right]$

$\mathrm{pinterp}≔\mathrm{CurveFitting}:-\mathrm{PolynomialInterpolation}\left(\mathrm{nodes}\,\mathrm{values}\,t\,\mathrm{form}=\mathrm{Lagrange}\right)$

$\mathrm{pinterp}≔\frac{9\left(t+\frac{1}{3}\right)\left(t-\frac{1}{3}\right)\left(t-1\right)}{16}+\frac{27\left(t+1\right)\left(t-\frac{1}{3}\right)\left(t-1\right)}{16}+\frac{27\left(t+1\right)\left(t+\frac{1}{3}\right)\left(t-1\right)}{16}+\frac{9\left(t+1\right)\left(t+\frac{1}{3}\right)\left(t-\frac{1}{3}\right)}{16}$

$\mathrm{normal}\left(\frac{\mathrm{pinterp}}{P:-\mathrm{Value}\left(t\right)\left[1,1\right]}\right)$

$1$

$N≔4$

$N≔4$

$p≔\mathrm{add}\left(\frac{k^3}{k+1}\mathrm{BernsteinBasis}\left(k\,N\,0\,1\,x\right)\,k=0..N\right)$

$p≔\frac{\mathrm{BernsteinBasis}\left(1\,4\,0\,1\,x\right)}{2}+\frac{8\mathrm{BernsteinBasis}\left(2\,4\,0\,1\,x\right)}{3}+\frac{27\mathrm{BernsteinBasis}\left(3\,4\,0\,1\,x\right)}{4}+\frac{64\mathrm{BernsteinBasis}\left(4\,4\,0\,1\,x\right)}{5}$

$P≔\mathrm{convert}\left(p\,\mathrm{MatrixPolynomialObject}\,x\right)$

$P≔\mathrm{Record}\left(\mathrm{Value}={\mathrm{Default}}_{\mathrm{value}}\,\mathrm{Variable}=x\,\mathrm{Degree}=4\,\mathrm{Coefficient}=\mathrm{coe}\,\mathrm{Dimension}=\left[1\,1\right]\,\mathrm{Basis}=\mathrm{BernsteinBasis}\,\mathrm{BasisParameters}=\left[4\,0\,1\right]\,\mathrm{IsMonic}=\mathrm{mon}\,\mathrm{OutputOptions}=\left[\mathrm{shape}=\left[\right]\,\mathrm{storage}=\mathrm{rectangular}\,\mathrm{order}=\mathrm{Fortran\_order}\,\mathrm{fill}=0\,\mathrm{attributes}=\left[\right]\right]\right)$

$\mathrm{type}\left(P\,\mathrm{MatrixPolynomialObject}\right)$

$\mathrm{true}$

$\mathrm{Digits}≔\mathrm{trunc}\left(\mathrm{evalhf}\left(\mathrm{Digits}\right)\right)$

$\mathrm{Digits}≔15$

$P:-\mathrm{Value}\left(0.3\right)$

$\left[\begin{array}{c}1.52538000000000\end{array}\right]$

$\mathrm{C0},\mathrm{C1}≔\mathrm{LinearAlgebra}\left[\mathrm{CompanionMatrix}\right]\left(P\,x\right)$

$\mathrm{C0},\mathrm{C1}≔\left[\begin{array}{cccc}0& 0& 0& 0\\ 1& 0& 0& -\frac{1}{2}\\ 0& 1& 0& -\frac{8}{3}\\ 0& 0& 1& -\frac{27}{4}\end{array}\right],\left[\begin{array}{cccc}4& 0& 0& 0\\ 1& \frac{3}{2}& 0& -\frac{1}{2}\\ 0& 1& \frac{2}{3}& -\frac{8}{3}\\ 0& 0& 1& -\frac{71}{20}\end{array}\right]$

$E≔\mathrm{LinearAlgebra}\left[\mathrm{Eigenvalues}\right]\left(\mathrm{evalf}\left(\mathrm{C0}\right)\,\mathrm{evalf}\left(\mathrm{C1}\right)\right)$

$E≔\left[\begin{array}{c}10.0659400730377+0.\mathrm{I}\\ \mathrm{-4.86159366800340}+0.\mathrm{I}\\ \mathrm{-0.204346405034140}+0.\mathrm{I}\\ 0.+0.\mathrm{I}\end{array}\right]$

$\mathrm{map}\left(t\→{P:-\mathrm{Value}\left(t\right)}_{1\,1}\,E\right)$

$\left[\begin{array}{c}\mathrm{-9.36779542826116}\times {10}^{\mathrm{-11}}\\ \mathrm{-1.00044417195022}\times {10}^{\mathrm{-11}}\\ \mathrm{-1.11022302462516}\times {10}^{\mathrm{-16}}\\ 0.\end{array}\right]$

$\mathrm{A0},\mathrm{A1},\mathrm{A2}≔\mathrm{LinearAlgebra}\left[\mathrm{RandomMatrix}\right]\left(3\,3\right),\mathrm{LinearAlgebra}\left[\mathrm{RandomMatrix}\right]\left(3\,3\right),\mathrm{LinearAlgebra}\left[\mathrm{RandomMatrix}\right]\left(3\,3\right)$

$\mathrm{A0},\mathrm{A1},\mathrm{A2}≔\left[\begin{array}{ccc}\mathrm{-74}& 8& 29\\ \mathrm{-4}& 69& 44\\ 27& 99& 92\end{array}\right],\left[\begin{array}{ccc}\mathrm{-98}& 27& \mathrm{-72}\\ \mathrm{-77}& \mathrm{-93}& \mathrm{-2}\\ 57& \mathrm{-76}& \mathrm{-32}\end{array}\right],\left[\begin{array}{ccc}\mathrm{-9}& 45& \mathrm{-18}\\ \mathrm{-50}& \mathrm{-81}& 87\\ \mathrm{-22}& \mathrm{-38}& 33\end{array}\right]$

$\mathrm{nodes}≔\left[-1\,0\,1\right]$

$\mathrm{nodes}≔\left[\mathrm{-1}\,0\,1\right]$

$P≔\mathrm{convert}\left(\left[\mathrm{A0}\,\mathrm{A1}\,\mathrm{A2}\right]\,\mathrm{MatrixPolynomialObject}\,\mathrm{nodes}\right)$

$P≔\mathrm{Record}\left(\mathrm{Value}={\mathrm{Default}}_{\mathrm{value}}\,\mathrm{Variable}=\mathrm{\_X4}\,\mathrm{Degree}=2\,\mathrm{Coefficient}=\mathrm{coe}\,\mathrm{Dimension}=\left[3\,3\right]\,\mathrm{Basis}=\mathrm{LagrangeBasis}\,\mathrm{BasisParameters}=\left[\left[\mathrm{-1}\,0\,1\right]\right]\,\mathrm{IsMonic}=\mathrm{mon}\,\mathrm{OutputOptions}=\left[\mathrm{shape}=\left[\right]\,\mathrm{storage}=\mathrm{rectangular}\,\mathrm{order}=\mathrm{Fortran\_order}\,\mathrm{fill}=0\,\mathrm{attributes}=\left[\right]\right]\right)$

$P:-\mathrm{Value}\left(0.3\right)$

$\left[\begin{array}{ccc}\mathrm{-83.1649999999999}& 32.5050000000000& \mathrm{-72.0749999999999}\\ \mathrm{-79.3999999999999}& \mathrm{-107.670000000000}& 10.5250000000000\\ 44.7449999999999& \mathrm{-86.9649999999999}& \mathrm{-32.3450000000000}\end{array}\right]$

$\mathrm{C0},\mathrm{C1}≔\mathrm{LinearAlgebra}\left[\mathrm{CompanionMatrix}\right]\left(P\,\mathrm{dummy}\right)$

$\mathrm{C0},\mathrm{C1}≔\begin{array}{c}\left[\begin{array}{ccccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0& 74& \dots \\ 0& 1& 0& 0& 0& 0& 0& 0& 0& \mathrm{-8}& \dots \\ 0& 0& 1& 0& 0& 0& 0& 0& 0& \mathrm{-29}& \dots \\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 98& \dots \\ 0& 0& 0& 0& 0& 0& 0& 0& 0& \mathrm{-27}& \dots \\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 72& \dots \\ 0& 0& 0& 0& 0& 0& \mathrm{-1}& 0& 0& 9& \dots \\ 0& 0& 0& 0& 0& 0& 0& \mathrm{-1}& 0& \mathrm{-45}& \dots \\ 0& 0& 0& 0& 0& 0& 0& 0& \mathrm{-1}& 18& \dots \\ \frac{1}{2}& 0& 0& \mathrm{-1}& 0& 0& \frac{1}{2}& 0& 0& 0& \dots \\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \end{array}\right]\\ \hfill \text{12 × 12 Matrix}\end{array},\begin{array}{c}\left[\begin{array}{ccccccccccc}\mathrm{-1}& 0& 0& 0& 0& 0& 0& 0& 0& 0& \dots \\ 0& \mathrm{-1}& 0& 0& 0& 0& 0& 0& 0& 0& \dots \\ 0& 0& \mathrm{-1}& 0& 0& 0& 0& 0& 0& 0& \dots \\ 0& 0& 0& \mathrm{-1}& 0& 0& 0& 0& 0& 0& \dots \\ 0& 0& 0& 0& \mathrm{-1}& 0& 0& 0& 0& 0& \dots \\ 0& 0& 0& 0& 0& \mathrm{-1}& 0& 0& 0& 0& \dots \\ 0& 0& 0& 0& 0& 0& \mathrm{-1}& 0& 0& 0& \dots \\ 0& 0& 0& 0& 0& 0& 0& \mathrm{-1}& 0& 0& \dots \\ 0& 0& 0& 0& 0& 0& 0& 0& \mathrm{-1}& 0& \dots \\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \dots \\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \end{array}\right]\\ \hfill \text{12 × 12 Matrix}\end{array}$