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+//page 304

+//Example 8.27

+clc;

+clear;

+close;

+disp('Unitary and orthogonal matrices');

+//part a

+disp('A = ');

+disp('[c]');

+disp('A is orthogonal if c = +1 or -1');

+disp('A is unitary if absolute value of c is 1, i.e. |c| = 1');

+disp('-------------------------------------------------');

+//part b

+disp('A = ');

+disp('a    b');

+disp('c    d');

+disp('A is orthogonal if, ');

+disp('A'' = inv(A)');

+disp('inv(A) = 1/(ad - bc) * X');

+disp('where X = ');

+disp(' d    -b');

+disp('-c     a');

+disp('Determinant of orthogonal matrices is +1 or -1');

+disp('So A is orthogonal if,');

+disp(' a   b');

+disp('-b   a');

+disp('or');

+disp('a    b');

+disp('b   -a');

+disp('where, a^2 + b^2 = 1');

+//part d

+disp('A is unitary if,');

+disp('A'' = inv(A)');

+disp('inv(A) = 1/(ad - bc) * X');

+disp('where X = ');

+disp(' d    -b');

+disp('-c     a');

+disp('Determinant of unitary matrices is +1 or -1');

+disp('So, A is unitary if,');

+disp('A = ');

+disp('a                                b');

+disp('-(e^i*x)*b_bar       (e^i*x)*a_bar');

+disp('A = ');

+disp('1    0                *            a         b');

+disp('0    e^(i*x)                    -b_bar     a_bar');

+disp('where x ia real number, and a,b are complex nos.');

+disp('|a|^2 + |b|^2 = 1');

+disp('-----------------------------------');

+//part c

+disp('A = ');

+disp('cos(thetha)         -sin(thetha)');

+disp('sin(thetha)          cos(thetha)');

+disp('A is orthogonal.');

+disp('If thetha is real, then A is unitary.');

+//end