(*^
::[	Information =

	"This is a Mathematica Notebook file.  It contains ASCII text, and can be
	transferred by email, ftp, or other text-file transfer utility.  It should
	be read or edited using a copy of Mathematica or MathReader.  If you 
	received this as email, use your mail application or copy/paste to save 
	everything from the line containing (*^ down to the line containing ^*)
	into a plain text file.  On some systems you may have to give the file a 
	name ending with ".ma" to allow Mathematica to recognize it as a Notebook.
	The line below identifies what version of Mathematica created this file,
	but it can be opened using any other version as well.";

	FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2";

	MacintoshStandardFontEncoding; 
	
	fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8,  24, "Times"; 
	fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6,  18, "Times"; 
	fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6,  14, "Times"; 
	fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20,  18, "Times"; 
	fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15,  14, "Times"; 
	fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12,  12, "Times"; 
	fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  14, "Times"; 
	fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; 
	fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5,  12, "Courier"; 
	fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5,  12, "Courier"; 
	fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5,  12, "Courier"; 
	fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5,  12, "Courier"; 
	fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5,  12, "Courier"; 
	fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, groupLikeGraphics, M7, l34, w405, h197,  12, "Courier"; 
	fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic,  10, "Geneva"; 
	fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = leftheader, inactive, L2,  12, "Times"; 
	fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7,  12, "Times"; 
	fontset = leftfooter, inactive, L2,  12, "Times"; 
	fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; 
	fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	paletteColors = 128; currentKernel; 
]
:[font = section; inactive; Cclosed; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Example 1: Rotations: 
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = text; inactive; preserveAspect]
Create a 2-D rotation matrix (rot[theta]) which is a function of theta.
Display this using MatrixForm.
;[s]
7:0,0;30,1;40,0;65,1;70,0;91,1;101,0;103,-1;
2:4,16,12,Times,0,14,0,0,0;3,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect]
rot[theta_]= {{Cos[theta],Sin[theta]},{-Sin[theta],Cos[theta]}};

rot[theta] //MatrixForm
:[font = text; inactive; preserveAspect]
Compute the inverse, and define this to be (roti[theta]).
Display this using MatrixForm.
;[s]
5:0,0;44,1;55,0;77,1;87,0;89,-1;
2:3,16,12,Times,0,14,0,0,0;2,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect]
roti[theta_] = Inverse[ rot[theta] ] //Simplify;

roti[theta] //MatrixForm
:[font = text; inactive; preserveAspect]
Verify rot.roti = unit matrix. 
;[s]
5:0,0;7,1;10,0;11,1;15,0;32,-1;
2:3,16,12,Times,0,14,0,0,0;2,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect]
rot[theta].roti[theta] //Simplify //MatrixForm
:[font = text; inactive; preserveAspect]
What is the relation between rot and roti. 

Now compute the Det of rot. 
;[s]
9:0,0;29,1;32,0;37,1;41,0;61,1;64,0;68,1;71,0;74,-1;
2:5,16,12,Times,0,14,0,0,0;4,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect]
Det[ rot[theta] ] //Simplify
:[font = text; inactive; preserveAspect]
We can obtain the Eigenvalues, Eigenvectors, or the entire Eigensystem using built-in Mathematic functions. 
;[s]
7:0,0;18,1;29,0;31,1;43,0;59,1;70,0;109,-1;
2:4,16,12,Times,0,14,0,0,0;3,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect]
Eigenvalues[ rot[theta] ] //Simplify
:[font = input; preserveAspect]
Eigenvectors[ rot[theta] ] //Simplify
:[font = input; preserveAspect]
Eigensystem[ rot[theta] ] //Simplify
:[font = text; inactive; preserveAspect]
Now, let's verify that a rotation does not alter the length of a vector. 
Let's define an arbitrary vector vec1=Array[v,2].
:[font = input; preserveAspect]
vec1=Array[v,2]
:[font = text; inactive; preserveAspect]
The length is: length1= Sqrt[vec1.vec1]
:[font = input; preserveAspect]
length1= Sqrt[vec1.vec1]
:[font = text; inactive; preserveAspect]
Define vec2= rot[theta].vec1
:[font = input; preserveAspect]
vec2= rot[theta].vec1
:[font = text; inactive; preserveAspect]
Then, the second length is: length2= Sqrt[vec2.vec2]
:[font = input; preserveAspect]
length2= Sqrt[vec2.vec2] //Simplify
:[font = text; inactive; preserveAspect]
Verify the length is invariant under a rotation. 
:[font = input; preserveAspect]
length1 == length2
:[font = subsection; inactive; preserveAspect; cellOutline; fontColorRed = 65535]
Exercise: Try generating the eigensystem this the old-fashioned way. 
:[font = subsection; inactive; preserveAspect; cellOutline; fontColorRed = 65535; endGroup]
Exercise: Verify the eigenvectors are orthonormal. 
:[font = section; inactive; Cclosed; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Problem 2: Boosts: 
Repeat the above exercise for rotations. 
Don't forget you will need a metric tensor g={1,-1} for this non-Euclidian space. 
;[s]
2:0,0;19,1;145,-1;
2:1,19,14,Times,1,18,0,0,0;1,16,12,Times,1,14,0,0,0;
:[font = text; inactive; preserveAspect]
We can think of a boost as a rotation by a complex angle. This will be a function of hyperbolic functions. 
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = text; inactive; preserveAspect]
Note, since we are in a non-Euclidian space, we will need to define a metric, g. 
:[font = input; preserveAspect]
g=DiagonalMatrix[{1,-1}];

g //MatrixForm
:[font = text; inactive; preserveAspect]
Create a 2-D boost matrix (boost[psi]) which is a function of psi.
Display this using MatrixForm.
;[s]
7:0,0;27,1;37,0;62,1;65,0;86,1;96,0;98,-1;
2:4,16,12,Times,0,14,0,0,0;3,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect]
boost[psi_]= {{Cosh[psi],Sinh[psi]},{Sinh[psi],Cosh[psi]}};

boost[psi] //MatrixForm
:[font = text; inactive; preserveAspect; plain; italic; fontColorRed = 65535]
In the following, I've left my comments, but deleted the input. 
:[font = text; inactive; preserveAspect]
Compute the inverse, and define this to be (boosti[psi]).
Display this using MatrixForm.
;[s]
5:0,0;44,1;55,0;77,1;87,0;89,-1;
2:3,16,12,Times,0,14,0,0,0;2,15,11,Courier,1,14,0,0,0;
:[font = text; inactive; preserveAspect]
Verify boost.boosti = unit matrix. 
;[s]
5:0,0;7,1;12,0;13,1;19,0;36,-1;
2:3,16,12,Times,0,14,0,0,0;2,15,11,Courier,1,14,0,0,0;
:[font = text; inactive; preserveAspect]
What is the relation between boost and boosti. 

Now compute the Det of boost. 
;[s]
9:0,0;29,1;34,0;39,1;45,0;65,1;68,0;72,1;77,0;80,-1;
2:5,16,12,Times,0,14,0,0,0;4,15,11,Courier,1,14,0,0,0;
:[font = text; inactive; preserveAspect]
We can obtain the Eigenvalues, Eigenvectors, or the entire Eigensystem using built-in Mathematic functions. 
;[s]
7:0,0;18,1;29,0;31,1;43,0;59,1;70,0;109,-1;
2:4,16,12,Times,0,14,0,0,0;3,15,11,Courier,1,14,0,0,0;
:[font = text; inactive; preserveAspect]
Now, let's verify that a rotation does not alter the length of a vector. 
Let's define an arbitrary vector vec1=Array[v,2].
:[font = text; inactive; preserveAspect]
The length is: length1= Sqrt[vec1.g.vec1]
Note, we must include the metric, g, here. 
;[s]
3:0,0;42,1;85,0;86,-1;
2:2,16,12,Times,0,14,0,0,0;1,16,12,Times,0,14,65535,0,0;
:[font = text; inactive; preserveAspect]
Define vec2= boost[psi].vec1
:[font = text; inactive; preserveAspect]
Then, the second length is: length2= Sqrt[vec2.g.vec2]
:[font = text; inactive; preserveAspect; endGroup]
Verify the length is invariant under a rotation. 
:[font = section; inactive; Cclosed; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Example 3:  Dihedral-3 Group
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = subsection; inactive; preserveAspect]
Find the matrix representation
:[font = text; inactive; preserveAspect]
We now consider the dihedral 3-group. This is can be thought of as the group that describes the symmetry of a equilateral triange.  A 2-D matrix representation is given by the following.  

First, we have a rotation by 1/3 of a revolution: 
:[font = input; preserveAspect]
r={{ Cos[th], -Sin[th] }
  ,{ Sin[th],  Cos[th]}}  /.{th->2 Pi/3};
  
r //MatrixForm
:[font = text; inactive; preserveAspect]
Next, we have a flip operation. This flips the x-axis, while leaving the y-axis unchanged. 
:[font = input; preserveAspect]
f= DiagonalMatrix[{-1,1}];

f //MatrixForm
:[font = text; inactive; preserveAspect]
We'll also need the 2-D identity matrix:
:[font = input; preserveAspect]
id= DiagonalMatrix[{1,1}];

id //MatrixForm
:[font = subsection; inactive; preserveAspect]
Find the order of the r and f sub-groups.
:[font = text; inactive; preserveAspect]
r and f form cyclic groups separately. Let's start by finding the order of these groups.  We find it convenient to use MatrixPower
:[font = input; preserveAspect; startGroup]
?MatrixPower
:[font = print; inactive; preserveAspect; endGroup]
MatrixPower[mat, n] gives the nth matrix power of mat.
:[font = input; preserveAspect]
Table[ MatrixPower[f,n] ,{n,1,2}] //N //MatrixForm
:[font = text; inactive; preserveAspect]
Notice that f^2 = id, the identity matrix. Therefore, f represents a sub-group of 2 elements, {1,f} since f^2=1. 
Let's do the same for r. 
:[font = input; preserveAspect]
Table[ MatrixPower[r,n] ,{n,1,3}] //N[#,3]&  //MatrixForm
:[font = text; inactive; preserveAspect]
Notice that r^3 = id, the identity matrix. Therefore, r represents a sub-group of 3 elements, {1,r,r^2} since r^3=1.
:[font = subsection; inactive; preserveAspect; startGroup]
Do r & f commute?
:[font = text; inactive; preserveAspect]
With Mathematica, the easiest way is to try it:
;[s]
3:0,0;5,1;16,0;48,-1;
2:2,16,12,Times,0,14,0,0,0;1,16,12,Times,2,14,0,0,0;
:[font = input; preserveAspect]
r.f //MatrixForm
:[font = input; preserveAspect]
f.r //MatrixForm
:[font = input; preserveAspect; endGroup]
r.f-f.r //MatrixForm
:[font = text; inactive; preserveAspect]
Nope, they don't commute. 
:[font = subsection; inactive; preserveAspect]
Combine G1={1,r,r^2} and G2={1,f}
:[font = text; inactive; preserveAspect]
Let's make a multiplication table for this group.  We start with the sub-groups we have identified, group1, group2. 
:[font = input; preserveAspect]
group1={id,r,r.r};
group2={id,f};
:[font = text; inactive; preserveAspect]
We now use Outer to multiply each element of group1 by each element of group2.  
(Note, we use the form Outer[ , , ,1] to tell Outer to use the upper level in extracting element of group1 and group2. To see what I mean, try it without the ",1]".)
:[font = text; inactive; preserveAspect]
To better understand the output of Outer, I construct this dummy example. 
:[font = input; preserveAspect]
Outer[Dot, {1,"r","r^2"},{1,"f"},1] //MatrixForm
:[font = input; preserveAspect]
Outer[Dot, group1,group2,1] //N[#,3]& //MatrixForm
:[font = text; inactive; preserveAspect]
This multiplication table gives all the elements of the group. 
:[font = text; inactive; preserveAspect]
Prove this, either by trial and error, or by analtyical methods.
For example, an arbitrary string of matrices yields:
:[font = input; preserveAspect]
f.r.f.f.r //N[#,3]& //MatrixForm
:[font = text; inactive; preserveAspect]
which we identify as the {2,2} element of the multiplication table, and therefore, this combination should be equal to r.f. 
:[font = input; preserveAspect; endGroup]
r.f == f.r.f.f.r
:[font = section; inactive; Cclosed; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Problem 4:  Dihedral-4 Group
Repeat the above exercise, but modify this for the Dihedral 4 group. 
;[s]
2:0,0;29,1;99,-1;
2:1,19,14,Times,1,18,0,0,0;1,16,12,Times,1,14,0,0,0;
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = input; preserveAspect]
We now consider the dihedral 4-group. This is can be thought of as the group that describes the symmetry of a square.  
;[s]
1:0,1;120,-1;
2:0,12,10,Courier,1,12,0,0,0;1,15,12,Times,0,14,0,0,0;
:[font = input; preserveAspect; fontColorRed = 65535; endGroup]
Copy from Example 3, and modify. 
;[s]
1:0,1;34,-1;
2:0,12,10,Courier,1,12,65535,0,0;1,15,12,Times,0,14,65535,0,0;
:[font = section; inactive; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981]
Optional Problem 5:  
Repeat the above exercise, but for other groups, eg., O(n), or SU(n), ...
;[s]
2:0,0;22,1;96,-1;
2:1,19,14,Times,1,18,0,0,0;1,16,12,Times,1,14,0,0,0;
^*)