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matrices : THEORY
BEGIN
IMPORTING vect2D,vect3D
Mat3 : TYPE = ARRAY[[below(3),below(3)]->real]
Mat2 : TYPE = {m:Mat3|m(2,0)=0 AND m(2,1)=0 AND m(2,2)=1}
m,n,l : VAR Mat3
v,w : VAR Vect3
x00,x01,x02,
x10,x11,x12,
x20,x21,x22,
k : VAR real
mat3(x00,x01,x02,x10,x11,x12,x20,x21,x22): Mat3 =
LAMBDA(x,y:below(3)):
IF (x=0 AND y=0) THEN x00
ELSIF (x=0 AND y=1) THEN x01
ELSIF (x=0 AND y=2) THEN x02
ELSIF (x=1 AND y=0) THEN x10
ELSIF (x=1 AND y=1) THEN x11
ELSIF (x=1 AND y=2) THEN x12
ELSIF (x=2 AND y=0) THEN x20
ELSIF (x=2 AND y=1) THEN x21
ELSIF (x=2 AND y=2) THEN x22
ELSE 0
ENDIF
putAt(m:Mat3, i, j:below(3),r:real): Mat3 = m WITH [(i,j) := r]
id: Mat3 = mat3(1,0,0,
0,1,0,
0,0,1)
row(m:Mat3, i:below(3)): Vect3 = vect3(m(i,0),m(i,1),m(i,2))
col(m:Mat3, i:below(3)): Vect3 = vect3(m(0,i),m(1,i),m(2,i))
% ------- Determinant
det(m): real = m(0,0)*m(1,1)*m(2,2) +
m(0,1)*m(1,2)*m(2,0) +
m(0,2)*m(1,0)*m(2,1) -
m(0,2)*m(1,1)*m(2,0) -
m(0,0)*m(1,2)*m(2,1) -
m(0,1)*m(1,0)*m(2,2)
singular?(m): bool = (det(m) = 0)
NonSingular : TYPE = {m: Mat3 | NOT singular?(m)}
ns : VAR NonSingular
det_id : LEMMA det(id) = 1
% ------- Transposed
T(m): Mat3 = mat3(m(0,0),m(1,0),m(2,0),
m(0,1),m(1,1),m(2,1),
m(0,2),m(1,2),m(2,2))
det_T: LEMMA det(m) = det(T(m));
% ------ Products
*(k,m): Mat3 = mat3(k*m(0,0),k*m(0,1),k*m(0,2),
k*m(1,0),k*m(1,1),k*m(1,2),
k*m(2,0),k*m(2,1),k*m(2,2));
*(v,m): Vect3 = vect3(v*col(m,0),v*col(m,1),v*col(m,2));
*(m,v): Vect3 = vect3(row(m,0)*v,row(m,1)*v,row(m,2)*v);
*(m,n): Mat3 = mat3(row(m,0)*col(n,0),row(m,0)*col(n,1),row(m,0)*col(n,2),
row(m,1)*col(n,0),row(m,1)*col(n,1),row(m,1)*col(n,2),
row(m,2)*col(n,0),row(m,2)*col(n,1),row(m,2)*col(n,2))
id_left1 : LEMMA id*m = m
id_right1: LEMMA m*id = m
id_left2 : LEMMA id*v = v
id_right2: LEMMA v*id = v
associativity_1 : LEMMA (m*n)*l = m*(n*l)
associativity_2 : LEMMA (m*n)*v = m*(n*v)
associativity_3 : LEMMA (v*m)*n = v*(m*n)
distribution_minus : LEMMA m*(v-w) = (m*v)-(m*w)
distribution_add : LEMMA m*(v+w) = (m*v)+(m*w)
% ------ Mat2
mat2(x00,x01,x10,x11): Mat2 = mat3(x00,x01,0,x10,x11,0,0,0,1)
det_mat2 : LEMMA det(mat2(x00,x01,x10,x11)) = x00*x11 - x01*x10
cofactor_ij(m:Mat3, i,j:below(3)): Mat3 = LAMBDA(a,b:below(3)):
IF (a=i AND b=j) THEN 1
ELSIF (a=i OR b=j) THEN 0
ELSE m(a,b)
ENDIF
cofactor(m:Mat3): Mat3 = LAMBDA(i,j:below(3)): det(cofactor_ij(m,i,j))
% ------ Inverse
inv(ns): Mat3 = (1/det(ns))*T(cofactor(ns))
inv_id : LEMMA inv(id) = id
inv_left : LEMMA inv(ns)*ns = id
inv_right : LEMMA ns*inv(ns) = id
% ------ pp: Mat3 -> (Vect3,Vect3,Vect3)
pp(m):[Vect3,Vect3,Vect3]= (row(m,0),row(m,1),row(m,2))
% ------ System of equations
% ------ Ax=B
sol_eqs(A:NonSingular,B:Vect3):Vect3 = inv(A)*B
sol_eqs_correctness : LEMMA ns*sol_eqs(ns,v) = v
% ------ 2 equations 2 variables
system2(v,u:Vect2): MACRO Mat2 = mat2(x(v),y(v),x(u),y(u))
% ------ 3 equations 3 variables
system3(v,u,w:Vect3): MACRO Mat3 =
mat3(x(v),y(v),z(v),x(u),y(u),z(u),x(w),y(w),z(w))
END matrices
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