(*^
::[	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,  12, "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, preserveAspect, groupLikeGraphics, M7, l34, w282, h287,  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; automaticGrouping; currentKernel; 
]
:[font = input; initialization; preserveAspect]
*)
$Post:=If[MatrixQ[#],MatrixForm[#],#]&
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 3.4.1
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[If[i>j,1,0],{i,4},{j,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0}, {1, 0, 0, 0}, {1, 1, 0, 0}, 
 
   {1, 1, 1, 0}}]
;[o]
0   0   0   0

1   0   0   0

1   1   0   0

1   1   1   0
:[font = input; initialization; preserveAspect; startGroup]
*)
Id[4]=IdentityMatrix[4];
p=Det[x Id[4]-A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
x^4
;[o]
 4
x
:[font = input; initialization; preserveAspect; startGroup]
*)
lambda=x/.Solve[p==0,x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0}
;[o]
{0, 0, 0, 0}
:[font = input; initialization; preserveAspect; startGroup]
*)
S=NullSpace[lambda[[1]]Id[4]-A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 1}}]
;[o]
0   0   0   1
:[font = input; initialization; preserveAspect; startGroup]
*)
M=NullSpace[S]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 1, 0}, {0, 1, 0, 0}, {1, 0, 0, 0}}]
;[o]
0   0   1   0

0   1   0   0

1   0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Join[S,M]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 1}, {0, 0, 1, 0}, {0, 1, 0, 0}, 
 
   {1, 0, 0, 0}}]
;[o]
0   0   0   1

0   0   1   0

0   1   0   0

1   0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].A.P
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 1, 1, 1}, {0, 0, 1, 1}, {0, 0, 0, 1}, 
 
   {0, 0, 0, 0}}]
;[o]
0   1   1   1

0   0   1   1

0   0   0   1

0   0   0   0
:[font = input; initialization; preserveAspect]
*)
Exercise 3.4.2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[Mod[i+j,2],{i,4},{j,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 1, 0, 1}, {1, 0, 1, 0}, {0, 1, 0, 1}, 
 
   {1, 0, 1, 0}}]
;[o]
0   1   0   1

1   0   1   0

0   1   0   1

1   0   1   0
:[font = input; initialization; preserveAspect; startGroup]
*)
p=Det[x Id[4]-A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
-4*x^2 + x^4
;[o]
    2    4
-4 x  + x
:[font = input; initialization; preserveAspect; startGroup]
*)
lambda=x/.Solve[p==0,x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-2, 0, 0, 2}
;[o]
{-2, 0, 0, 2}
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[S]
Do[S[i]=NullSpace[lambda[[i]]Id[4]-A],{i,4}]
{S[1],S[2],S[4]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{{-1, 1, -1, 1}}, {{0, -1, 0, 1}, {-1, 0, 1, 0}}, 
 
  {{1, 1, 1, 1}}}
;[o]
{{{-1, 1, -1, 1}}, {{0, -1, 0, 1}, {-1, 0, 1, 0}}, 
 
  {{1, 1, 1, 1}}}
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Join[S[1],S[2],S[4]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1, 0, -1, 1}, {1, -1, 0, 1}, {-1, 0, 1, 1}, 
 
   {1, 1, 0, 1}}]
;[o]
-1   0    -1   1

1    -1   0    1

-1   0    1    1

1    1    0    1
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].A.P
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-2, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, 
 
   {0, 0, 0, 2}}]
;[o]
-2   0    0    0

0    0    0    0

0    0    0    0

0    0    0    2
:[font = input; initialization; preserveAspect]
*)
Exercise 3.4.3
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[If[i<=2,2,0],{i,4},{j,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 2, 2, 2}, {2, 2, 2, 2}, {0, 0, 0, 0}, 
 
   {0, 0, 0, 0}}]
;[o]
2   2   2   2

2   2   2   2

0   0   0   0

0   0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
p=Det[x Id[4]-A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
-4*x^3 + x^4
;[o]
    3    4
-4 x  + x
:[font = input; initialization; preserveAspect; startGroup]
*)
lambda=x/.Solve[p==0,x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 4}
;[o]
{0, 0, 0, 4}
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[S[i]=NullSpace[lambda[[i]]Id[4]-A],{i,4}]
{S[1],S[4]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{{-1, 0, 0, 1}, {-1, 0, 1, 0}, {-1, 1, 0, 0}}, 
 
  {{1, 1, 0, 0}}}
;[o]
{{{-1, 0, 0, 1}, {-1, 0, 1, 0}, {-1, 1, 0, 0}}, 
 
  {{1, 1, 0, 0}}}
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Join[S[1],S[4]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1, -1, -1, 1}, {0, 0, 1, 1}, {0, 1, 0, 0}, 
 
   {1, 0, 0, 0}}]
;[o]
-1   -1   -1   1

0    0    1    1

0    1    0    0

1    0    0    0
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].A.P
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, 
 
   {0, 0, 0, 4}}]
;[o]
0   0   0   0

0   0   0   0

0   0   0   0

0   0   0   4
:[font = input; initialization; preserveAspect]
*)
Exercise 3.4.4
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[If[i==j,0,1],{i,4},{j,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 1, 1, 1}, {1, 0, 1, 1}, {1, 1, 0, 1}, 
 
   {1, 1, 1, 0}}]
;[o]
0   1   1   1

1   0   1   1

1   1   0   1

1   1   1   0
:[font = input; initialization; preserveAspect; startGroup]
*)
p=Det[x Id[4]-A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
-3 - 8*x - 6*x^2 + x^4
;[o]
              2    4
-3 - 8 x - 6 x  + x
:[font = input; initialization; preserveAspect; startGroup]
*)
lambda=x/.Solve[p==0,x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-1, -1, -1, 3}
;[o]
{-1, -1, -1, 3}
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[S[i]=NullSpace[lambda[[i]]Id[4]-A],{i,4}]
{S[1],S[4]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{{-1, 0, 0, 1}, {-1, 0, 1, 0}, {-1, 1, 0, 0}}, 
 
  {{1, 1, 1, 1}}}
;[o]
{{{-1, 0, 0, 1}, {-1, 0, 1, 0}, {-1, 1, 0, 0}}, 
 
  {{1, 1, 1, 1}}}
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Join[S[1],S[4]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1, -1, -1, 1}, {0, 0, 1, 1}, {0, 1, 0, 1}, 
 
   {1, 0, 0, 1}}]
;[o]
-1   -1   -1   1

0    0    1    1

0    1    0    1

1    0    0    1
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].A.P
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1, 0, 0, 0}, {0, -1, 0, 0}, {0, 0, -1, 0}, 
 
   {0, 0, 0, 3}}]
;[o]
-1   0    0    0

0    -1   0    0

0    0    -1   0

0    0    0    3
^*)