ExampleMagmaCode5.mgm

// Examples on writing functions

Z := Integers();
Q := Rationals();
R := RealField(20);
P<x> := PolynomialRing(Q);

Matrix(Z, 3,2, [1,2,3,4,5,6]);
Matrix(Z, 2,3, [1,2,3,4,5,6]);
Matrix(Z, [ [1,2,3],[4,5,6] ]);

// We get errors if we don't give enough entries
Matrix(Z, 2,3, [1,2,3,4,5]);
Matrix(Z, [ [1,2,3],[4,5] ]);

// We can leave out the field/ring and let Magma decide
Matrix([ [1,2,3],[4,5,6] ]);

// Parent command gives information about an object,
// can tell us things like where coefficients live, etc.
Parent( Matrix([ [1,2,3],[4,5,6] ]) );
Parent( Matrix([ [1/1,2,3],[4,5,6] ]) );
Parent( Matrix(Q, [ [1,2,3],[4,5,6] ]) );

// Have some special commands for structured matrices
DiagonalMatrix(Q, [1,2,3,4,5];
DiagonalMatrix(Q, [1,2,3,4,5]);
UpperTriangularMatrix(Q, [1,2,3,4]);
UpperTriangularMatrix(Q, [1,2,3,4,5,6]);
SymmetricMatrix(Q, [1,2,3,4,5,6]);

// Vectors created just like matrices
Vector( [1,2,3,4]);
Vector(Q, [1,2,3,4]);
Vector(R, [1,2,3,4]);

A := Matrix( [ [1,4], [2,5], [3,6] ] );
A;
v := Vector([7,8,9]);

// Vectors are rows, multiply from left of a matrix.
A*v;
v*A;

// Can have polynomial entries:
A := Matrix( [ [x^2,2*x], [3*x,5*x^3], [4,6*x] ] );
A;
v*A;
v1 := Vector([2*x^2, 5*x]);
v2 := Vector([5,7]);
v1;
v2;
// This gives an error because Magma decided v1 has entries from P,
// v2 has entries from Z
(v1, v2);

// By making sure entries of v2 are viewed as polynomials, we can compute dot product
v1 := Vector([2*x^2, 5*x]);
v2 := Vector(P, [5,7]);
v1;
v2;
(v1, v2);



M1 := Matrix(2,3, [1,2,3,4,5,6]);
M2 := Matrix(3,5, [ i^2 : i in [1..15]]);
M2;
M1;
M1*M2;
Transpose(M2;
Transpose(M2);
M2;
EchelonForm(M2);
u := Vector(Q, [1,5,7]);


// We know that we can project a vector x onto the line through u
// using the formula ((u,x)/(u,u))*u.
function ProjU(x)
  return ((u,x) / (u,u))*u;
end function;

ProjU( Vector(Q, [13,5,3] ));

// This function takes a vector u, then builds & returns a function
function ProjFunc(u)
  function ProjU(x)
    return ((u,x) / (u,u))*u;
  end function;
  return ProjU;
end function;

f := ProjFunc( Vector(Q, [1,1,1]));
f( Vector(Q, [3,4,7]));

// Echelon Form depends on where the coefficients live,
// Magma will only do row operations involving coefficients structure
M2;
EchelonForm(M2);
M2 := Matrix(Q,3,5, [ i^2 : i in [1..15]]);
EchelonForm(M2);
// For integers, Magma won't multiply the row by (1/5),
// even though the result would have integer entries.
// This reduces to reduced echelon over the rational numbers.

// EchelonForm(M) returns E, B so that B*M = E
E,B := EchelonForm(M2);
E;
B;
B*M2 eq E;


M := Matrix(R, 2,2, [0,1,-1,0]);
M;
Eigenvalues(M);
M := Matrix(R, 2,2, [0,1,2,0]);
Eigenvalues(M);
M := Matrix(Q, 2,2, [0,1,2,0]);
Eigenvalues(M);
M := Matrix(R, 2,2, [0,1,2,0]);
M := Matrix(R, 2,2, [0,1,2,0]);
M;
EV := Eigenvalues(M);
EV;
EV[1];
lambda := EV[1][1];
lambda;
Eigenspace(M, lambda);
Eigenspaces(M);
Eigenspace;
M := Matrix(Q, 4,4, [0, -1, 0, -4, 1, 0, 3, 5, 4, 1, 2, 4, 6, 8, 0]);
M := Matrix(Q, 4,4, [0, -1, 0, -4, 1, 0, 3, 5,3, 4, 1, 2, 4, 6, 8, 0]);
M;
Eigenvalues(M);
M := DiagonalMatrix(Q, [1,1,3,3]);
M;
Eigenvalues(M);
Eigenspace(M, 3);
V := VectorSpace(Q,4);
V;
v1 := Vector(Q, [1,3,2,7]);
v2 := Vector(Q, [3,-1,0,1]);
v1 in V;
elt<V | 1,3,2,7> eq v1;
V![1,3,2,7] eq v1;
Normalize(v2);
u := (1/Norm(v2)) * v2;
u;
U := sub<V | v1, v2>;
U;
A := Matrix(Q, [[1,2,3], [2,3,4], [3,4,5]]);
A;
Inverse(A);
A^-1;
Determinant(A);