From 7f60ea012dd2524dae921a2a35adbf7ef21f2bb6 Mon Sep 17 00:00:00 2001
From: prashantsinalkar
Date: Tue, 10 Oct 2017 12:27:19 +0530
Subject: initial commit / add all books

---
 3293/CH8/EX8.23/Ex8_23.sce | 33 ++++++++++++++++++++++++++++
 3293/CH8/EX8.27/Ex8_27.sce | 55 ++++++++++++++++++++++++++++++++++++++++++++++
 2 files changed, 88 insertions(+)
 create mode 100644 3293/CH8/EX8.23/Ex8_23.sce
 create mode 100644 3293/CH8/EX8.27/Ex8_27.sce

(limited to '3293/CH8')

diff --git a/3293/CH8/EX8.23/Ex8_23.sce b/3293/CH8/EX8.23/Ex8_23.sce
new file mode 100644
index 000000000..ec9cd79f0
--- /dev/null
+++ b/3293/CH8/EX8.23/Ex8_23.sce
@@ -0,0 +1,33 @@
+//page 301
+//Example 8.23
+clc;
+clear;
+close;
+disp('Linear transformation from V into W i.e. T is:');
+disp('T(x1,x2,x3) = ');
+disp('0    -x3    x2');
+disp('x3     0   -x1');
+disp('-x2   x1    0');
+disp('Then, T maps V onto W');
+disp('And, putting:');
+disp('A = ');
+disp('0    -x3    x2');
+disp('x3     0   -x1');
+disp('-x2   x1    0');
+disp('B = ');
+disp('0    -y3    y2');
+disp('y3     0   -y1');
+disp('-y2   y1    0');
+disp('we get,');
+disp('tr(AB'') = x3*y3 + x2*y2 + x1*y1 + x3*y3 + x2*y2 + x1*y1');
+disp('tr(AB'') = 2*(x1*y1 + x2*y2 + x3*y3)');
+disp('Thus, (a|b) = (Ta|Tb)');
+disp('T is vector space isomorphism');
+disp('T contains the standard and orthonormal basis consisting of matrices A1,A2,A3');
+A1 = [0 0 0;0 0 -1;0 1 0];
+A2 = [0 0 1;0 0 0;-1 0 0];
+A3 = [0 -1 0;1 0 0;0 0 0];
+disp(A1,'A1 = ');
+disp(A2,'A2 = ');
+disp(A3,'A3 = ');
+//end
diff --git a/3293/CH8/EX8.27/Ex8_27.sce b/3293/CH8/EX8.27/Ex8_27.sce
new file mode 100644
index 000000000..a91dc1932
--- /dev/null
+++ b/3293/CH8/EX8.27/Ex8_27.sce
@@ -0,0 +1,55 @@
+//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
-- 
cgit