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A short demonstration of Axiom

An important thing: all objects in Axiom have a type. This enables us to give a simple demonstration of the Cayley-Hamilton theorem.

Let n equal 4. The semicolon at the end of the input tells Axiom not to display the result. Thus, only its type is shown:

axiom

n := 4;

Type: PositiveInteger

We define an abbreviation: let SM be the ring of quadratic $n\times n$ matrices with rational functions as entries:

axiom

SM ==> SquareMatrix(n, FRAC POLY INT)

Type: Void

Let M be a generic $4 \times 4$ matrix:

axiom

M: SM := matrix [[a[i,j] for j in 1..n] for i in 1..n]

$\label{eq1}\left[ \begin{array}{cccc} {a_{1, \: 1}}&{a_{1, \: 2}}&{a_{1, \: 3}}&{a_{1, \: 4}} \ {a_{2, \: 1}}&{a_{2, \: 2}}&{a_{2, \: 3}}&{a_{2, \: 4}} \ {a_{3, \: 1}}&{a_{3, \: 2}}&{a_{3, \: 3}}&{a_{3, \: 4}} \ {a_{4, \: 1}}&{a_{4, \: 2}}&{a_{4, \: 3}}&{a_{4, \: 4}}$

(1)

Type: SquareMatrix(4,Fraction(Polynomial(Integer)))

Compute the characteristis polynomial of 'M':

axiom

P := determinant (M - x * 1);

Type: Fraction(Polynomial(Integer))

We now interpret P as univariate polynomial in x, coefficients being from the $4 \times 4$ matrices. The double colon performs this coercion:

axiom

Q := P::UP(x, SM);

Type: UnivariatePolynomial(x,SquareMatrix(4,Fraction(Polynomial(Integer))))

Finally we evaluate this polynomial with the original matrix as argument. In Axiom you do not need to use parenthesis if the function takes only one argument:

axiom

Q M

$\label{eq2}\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(2)

Type: SquareMatrix(4,Fraction(Polynomial(Integer)))

Some matrix computations under TeXmacs.

Notice the hierarchical editing capabilities of TeXmacs.

Some more complicated computations:

axiom

)cl all

   All user variables and function definitions have been cleared.
Word := OrderedFreeMonoid(Symbol)

$\label{eq3}\hbox{\axiomType{OrderedFreeMonoid}\ } (\hbox{\axiomType{Symbol}\ })$

(3)

Type: Domain

axiom

poly:= XPR(Integer,Word)

$\label{eq4}\hbox{\axiomType{XPolynomialRing}\ } (\hbox{\axiomType{Integer}\ } , \hbox{\axiomType{OrderedFreeMonoid}\ } (\hbox{\axiomType{Symbol}\ }))$

(4)

Type: Domain

axiom

p:poly := 2 * x - 3 * y + 1

$\label{eq5}1 +{2 \ x}-{3 \ y}$

(5)

Type: XPolynomialRing(Integer,OrderedFreeMonoid(Symbol))

axiom

1 + 2x - 3y

   Cannot find a definition or applicable library operation named 2
      with argument type(s)
                                 Variable(x)

      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.
q:poly := 2 * x + 1

$\label{eq6}1 +{2 \ x}$

(6)

Type: XPolynomialRing(Integer,OrderedFreeMonoid(Symbol))

axiom

p + q

$\label{eq7}2 +{4 \ x}-{3 \ y}$

(7)

Type: XPolynomialRing(Integer,OrderedFreeMonoid(Symbol))

axiom

p * q

$\label{eq8}1 +{4 \ x}-{3 \ y}+{4 \ {x^2}}-{6 \ y \ x}$

(8)

Type: XPolynomialRing(Integer,OrderedFreeMonoid(Symbol))

axiom

(p +q)^2 -p^2 -q^2 - 2*p*q

$\label{eq9}-{6 \ x \ y}+{6 \ y \ x}$

(9)

Type: XPolynomialRing(Integer,OrderedFreeMonoid(Symbol))

axiom

M := SquareMatrix(2,Fraction Integer)

$\label{eq10}\hbox{\axiomType{SquareMatrix}\ } (2, \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }))$

(10)

Type: Domain

axiom

poly1:= XPR(M,Word)

$\label{eq11}\hbox{\axiomType{XPolynomialRing}\ } (\hbox{\axiomType{SquareMatrix}\ } (2, \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ })) , \hbox{\axiomType{OrderedFreeMonoid}\ } (\hbox{\axiomType{Symbol}\ }))$

(11)

Type: Domain

axiom

m1:M := matrix [[i*j**2 for i in 1..2] for j in 1..2]

$\label{eq12}\left[ \begin{array}{cc} 1 & 2 \ 4 & 8$

(12)

Type: SquareMatrix(2,Fraction(Integer))

axiom

m2:M := m1 - 5/4

$\label{eq13}\left[ \begin{array}{cc} -{1 \over 4}& 2 \ 4 &{{27}\over 4}$

(13)

Type: SquareMatrix(2,Fraction(Integer))

axiom

m3: M := m2**2

$\label{eq14}\left[ \begin{array}{cc} {{129}\over{16}}&{13} \ {26}&{{857}\over{16}}$

(14)

Type: SquareMatrix(2,Fraction(Integer))

axiom

pm:poly1   := m1*x + m2*y + m3*z - 2/3

$\label{eq15}\begin{array}{@{}l} \displaystyle {\left[ \begin{array}{cc} -{2 \over 3}& 0 \ 0 & -{2 \over 3}$

(15)

Type: XPolynomialRing(SquareMatrix(2,Fraction(Integer)),OrderedFreeMonoid(Symbol))

axiom

qm:poly1 := pm - m1*x

$\label{eq16}\begin{array}{@{}l} \displaystyle {\left[ \begin{array}{cc} -{2 \over 3}& 0 \ 0 & -{2 \over 3}$

(16)

Type: XPolynomialRing(SquareMatrix(2,Fraction(Integer)),OrderedFreeMonoid(Symbol))

axiom

qm**3

$\label{eq17}\begin{array}{@{}l} \displaystyle {\left[ \begin{array}{cc} -{8 \over{27}}& 0 \ 0 & -{8 \over{27}}$

(17)

Type: XPolynomialRing(SquareMatrix(2,Fraction(Integer)),OrderedFreeMonoid(Symbol))

Subject: Be Bold !!

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