Section9.58

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## 9.58 Operator

Given any ring R, the ring of the Integer-linear operators over R is called Operator(R). To create an operator over R, first create a basic operator using the operation operator, and then convert it to Operator(R) for the R you want.

We choose R to be the two by two matrices over the integers.

R := SQMATRIX(2, INT)

SquareMatrix(2, Integer)

Type: Domain

Create the operator tilde on R.

t := operator("tilde") :: OP(R)

tilde

Type: Operator SquareMatrix(2,Integer)

Since Operator is unexposed we must either package-call operations from it, or expose it explicitly. For convenience we will do the latter.

Expose Operator.

)set expose add constructor Operator
Operator is now explicitly exposed in frame G82322

To attach an evaluation function (from R to R) to an operator over R, use evaluate(op, f) where op is an operator over R and f is a function R -> R. This needs to be done only once when the operator is defined. Note that f must be Integer-linear (that is, f(ax+y) = a f(x) + f(y) for any integer a, and any x and y in R).

We now attach the transpose map to the above operator t.

evaluate(t, m +-> transpose m)

tilde

Type: Operator SquareMatrix(2,Integer)

Operators can be manipulated formally as in any ring: + is the pointwise addition and * is composition. Any element x of R can be converted to an operator op x over R, and the evaluation function of op x is left-multiplication by x.

Multiplying on the left by this matrix swaps the two rows.

s : R := matrix [ [0, 1], [1, 0] ]

[0 1 1 0]

Type: SquareMatrix(2,Integer)

Can you guess what is the action of the following operator?

rho := t * s

tilde[0 1 1 0]

Type: Operator SquareMatrix(2,Integer)

Hint: applying rho four times gives the identity, so rho**4-1 should return 0 when applied to any two by two matrix.

z := rho**4 - 1

-1+tilde[0 1 1 0]tilde[0 1 1 0]tilde[0 1 1 0]tilde[0 1 1 0]

Type: Operator SquareMatrix(2,Integer)

Now check with this matrix.

m:R := matrix [ [1, 2], [3, 4] ]

[1 2 3 4]

Type: SquareMatrix(2,Integer)
z m

[0 0 0 0]

Type: SquareMatrix(2,Integer)

As you have probably guessed by now, rho acts on matrices by rotating the elements clockwise.

rho m

[3 1 4 2]

Type: SquareMatrix(2,Integer)
rho rho m

[4 3 2 1]

Type: SquareMatrix(2,Integer)
(rho**3) m

[2 4 1 3]

Type: SquareMatrix(2,Integer)

Do the swapping of rows and transposition commute? We can check by computing their bracket.

b := t * s - s * t

-[0 1 1 0]tilde+tilde[0 1 1 0]

Type: Operator SquareMatrix(2,Integer)

Now apply it to m.

b m

[1-3 3-1]

Type: SquareMatrix(2,Integer)

Next we demonstrate how to define a differential operator on a polynomial ring.

This is the recursive definition of the n-th Legendre polynomial.

L n ==
n = 0 => 1
n = 1 => x
(2*n-1)/n * x * L(n-1) - (n-1)/n * L(n-2)
Void

Create the differential operator d dx on polynomials in x over the rational numbers.

dx := operator("D") :: OP(POLY FRAC INT)

D

Type: Operator Polynomial Fraction Integer

Now attach the map to it.

evaluate(dx, p +-> D(p, 'x))

D

Type: Operator Polynomial Fraction Integer

This is the differential equation satisfied by the n-th Legendre polynomial.

E n == (1 - x**2) * dx**2 - 2 * x * dx + n*(n+1)
Void

Now we verify this for n = 15. Here is the polynomial.

L 15

9694845 2048 x 15-35102025 2048 x 13+50702925 2048 x 11-37182145 2048 x 9+14549535 2048 x 7-2909907 2048 x 5+255255 2048 x 3-6435 2048 x

Type: Polynomial Fraction Integer

Here is the operator.

E 15

240-2 x D-(x 2-1)D 2

Type: Operator Polynomial Fraction Integer

Here is the evaluation.

(E 15)(L 15)

0

Type: Polynomial Fraction Integer
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