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Editor: test1 Time: 2015/02/06 18:02:02 GMT+0
Note:

removed:
-1 + 2x - 3y

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

changed:
-m3: M := m2**2
m3: M := m2^2

changed:
-qm**3
qm^3

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:

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n := 4;

Type: PositiveInteger?

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

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

Type: Void

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

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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':

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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:

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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:

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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 OldTeXmacs ?.

Notice the hierarchical editing capabilities of OldTeXmacs?.

Some more complicated computations:

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)cl all

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

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

(3)

Type: Type

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poly:= XPR(Integer,Word)

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

(4)

Type: Type

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

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