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:

n := 4;

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

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

Let M be a generic matrix:

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

(1)

Compute the characteristis polynomial of 'M':

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

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

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

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:

Q M

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Some matrix computations under TeXmacs.

Notice the hierarchical editing capabilities of TeXmacs.

Some more complicated computations:

)cl all

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

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poly:= XPR(Integer,Word)

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

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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

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p + q

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p * q

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

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M := SquareMatrix(2,Fraction Integer)

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poly1:= XPR(M,Word)

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

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m2:M := m1 - 5/4

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m3: M := m2**2

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

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qm:poly1 := pm - m1*x

(16)

qm**3

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