%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% this is an octave test file
% call it with the command
%
% $ octave octave_test.m
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A = [1,1;2,3]
det_A = det(A)
B = inv(A)
b = [4;5]
x = A\b

O = A*x - b




%%%%%%%%%%%%%%%%% exercise sheet III, exo 3
A=[3,2,-1;-2,-1,0;4,3,-1]
 M=[0, -1, -2;
       0, 0,   2;
       1, -1, -3]
 
 MI=inv(M)
 
 MI*A*M
 
 P=[ -1, 0.5, 1;
       -1, -1, 0;
       0, 0.5, 0]
 PI=inv(P)
 
 ERG= P*A*PI
 
 det(A)
 det (M)
 det(P)
 det(A')
 rank(A)
 lambda = eig(A)
 [V, lambda2] = eig(A)
 
%norm(A, 2)
traceA = trace(A)
%expm(A)
%logm(A)
%sqrtm(A)
%%%%%%%%%%%%%%% an example for lacking precision in calculation: result should be zero
V*lambda2*inv(V) - A


%%%%%%%%%%%%%%   III /  3
printf("sheet III, exo 2\n")
A = [1,0,1;1,1,2;0,1,1]
B = [1,0,1;0,1,0;0,0,1]
det_A = det(A)
det_B = det(B)
rank_A = rank(A)
rank_B = rank(B)

C = [1,1;1,-1]
D = [0,1;1,0]
det_C = det(C)
det_D = det(D)
rank_C = rank(C)
rank_D = rank(D)



%%%%%%%%%%%%%   III / 7
7_A = [1, 5, -1;2, -2, -2; 3, -3, 4]
7_B = [3, 7, -1, 1;2, -2, 1, 1; 1, 1, -4, 0; -1, 1, -2, 3]