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 opx over R, and the evaluation function of opx is left-multiplication by x.
Multiplying on the left by this matrix swaps the two rows.
s : R := matrix [ [0, 1], [1, 0] ]
| [0110] |
Type: SquareMatrix(2,Integer)
Can you guess what is the action of the following operator?
rho := t * s
| tilde[0110] |
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[0110]tilde[0110]tilde[0110]tilde[0110] |
Type: Operator SquareMatrix(2,Integer)
Now check with this matrix.
m:R := matrix [ [1, 2], [3, 4] ]
| [1234] |
Type: SquareMatrix(2,Integer)
z m
| [0000] |
Type: SquareMatrix(2,Integer)
As you have probably guessed by now, rho acts on matrices by rotating the elements clockwise.
rho m
| [3142] |
Type: SquareMatrix(2,Integer)
rho rho m
| [4321] |
Type: SquareMatrix(2,Integer)
(rho^3) m
| [2413] |
Type: SquareMatrix(2,Integer)
Do the swapping of rows and transposition commute? We can check by computing their bracket.
b := t * s - s * t
| -[0110]tilde+tilde[0110] |
Type: Operator SquareMatrix(2,Integer)
Now apply it to m.
b m
| [1-33-1] |
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 ddx 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
| 96948452048x15-351020252048x13+507029252048x11-371821452048x9+145495352048x7-29099072048x5+2552552048x3-64352048x |
Type: Polynomial Fraction Integer
Here is the operator.
E 15
| 240-2xD-(x2-1)D2 |
Type: Operator Polynomial Fraction Integer
Here is the evaluation.
(E 15)(L 15)
| 0 |
Type: Polynomial Fraction Integer