Re: matrices & polynomials in mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg27290] Re: [mg27273] matrices & polynomials in mathematica
• From: Tomas Garza <tgarza01 at prodigy.net.mx>
• Date: Sat, 17 Feb 2001 03:31:12 -0500 (EST)
• References: <200102160858.DAA13465@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Let your polynomial be f(v), and a the matrix you want to substitute into
the polynomial (I use a so as to avoid capital letters, which are bad form
in Mathematica). Obtain the coefficients of p using CoefficientList
CoefficientList[poly, var] gives a list of coefficients of powers of var in
poly, starting with power 0). Suppose, for example, that

In[1]:=
k = {c1, c2, c3, c4};

is the list of coefficients thus obtained (i.e., your polynomial is of
degree 3). Take, for example,

In[2]:=
a = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}};

and define, for an arbitrary matrix x,

In[3]:=
g[y_] := Dot[y, x]

Then

In[4]:=
NestList[g, x, Length[k] - 2]
Out[4]=
{x, x.x, x.x.x}

so that

In[5]:=
Rest[k].NestList[g, x, Length[k] - 2]
Out[5]=
c2 x + c3 x.x + c4 x.x.x

Then, in particular, for x = a all you have to do is add the first term:

In[6]:=
First[k]*IdentityMatrix[Dimensions[x][[1]]] + Rest[k].NestList[g, x,
Length[k] - 2]

I suppose you could subsume all this into a function:

polToMat[a_?MatrixQ, k_List] := (g[y_] := Dot[y, a];
First[k]*IdentityMatrix[Dimensions[a][[1]]] +
Rest[k].NestList[g, a, Length[k] - 2])

Example:

In[7]:=
polToMat[{{1, 2}, {3, 4}}, CoefficientList[1 + x + x^2, x]]
Out[7]=
{{9, 12}, {18, 27}}

Tomas Garza
Mexico City

----- Original Message -----
From: "news.tue.nl" <student at tue.nl>
To: mathgroup at smc.vnet.net
Subject: [mg27290] [mg27273] matrices & polynomials in mathematica

> Hi,
>
> I have the following problem.
> I have written a mathematica-module that automatically produces a certain
> polynomial as a result. But I want another module to use this first module
> and fill in a matrix in this polynomial.
> for example: f[x]=1+x+x^2
> But when I compute f[A] where A is a matrix, mathematica interprets 1 as a
> matrix filled with ones in stead of the identity-matrix. And A^2 is
> pointswise multiplication in stead of matrix-multiplication.
> Is there an option to let Mathematica know that the polynomial works on
> matrices?
>
> Chris
>
> PS. I am aware of the possibility to multiply matrices using . (dot) but
the
> problem is that the polynomial is the result of some computations and I
need
> to have mathematica interpret this polynomial on matrices automatically.
>
>
>
>
>

```

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