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  of the matrix polynomial ( 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).