(*^
::[	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"; 
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	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"; 
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	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"; 
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	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"; 
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	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 3.5.1
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[If[i==j,i+j-1,i+j+1],{i,5},{j,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 4, 5, 6, 7}, {4, 3, 6, 7, 8}, 
 
   {5, 6, 5, 8, 9}, {6, 7, 8, 7, 10}, {7, 8, 9, 10, 9}}]
;[o]
1    4    5    6    7

4    3    6    7    8

5    6    5    8    9

6    7    8    7    10

7    8    9    10   9
:[font = input; initialization; preserveAspect; startGroup]
*)
ES=Eigensystem[A];
lambda=ES[[1]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-2, -2, -2, (31 - 5*57^(1/2))/2, (31 + 5*57^(1/2))/2}
;[o]
             31 - 5 Sqrt[57]  31 + 5 Sqrt[57]
{-2, -2, -2, ---------------, ---------------}
                    2                2
:[font = input; initialization; preserveAspect; startGroup]
*)
Q=ES[[2]]//Together; Q//N
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{3., -4., 0, 0, 1.}, {2., -3., 0, 1., 0}, 
 
   {1., -2., 1., 0, 0}, {-1.318729304408843503, 
 
    -0.7390469783066326857, -0.1593646522044219153, 
 
    0.4203176738977889488, 1.}, 
 
   {0.5687293044088437121, 0.6765469783066327841, 
 
    0.784364652204421856, 0.8921823261022109281, 1.}}]
;[o]
3.          -4.         0           0           1.

2.          -3.         0           1.          0

1.          -2.         1.          0           0

-1.31873    -0.739047   -0.159365   0.420318    1.

0.568729    0.676547    0.784365    0.892182    1.
:[font = input; initialization; preserveAspect]
*)
We observe that there are 5 nonzero, linearly independent 
eigenvectors, so A is diagonalizable. (The mention 
"nonzero" from above could be considered redundant, since 
in general we only consider nonzero eigenvectors - the
zero vector being a trivial eigenvector for any eigenvalue
and any square matrix of matching size. However, if the
number of linearly independent eigenvectors does not match
the multiplicity of an eigenvalue, Mathematica inserts a
number of null eigenvectors up to that multiplicity.) 
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Q]; P//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{3, 2, 1, (3*(-491 + 65*57^(1/2)))/
 
     (-2589 + 343*57^(1/2)), 
 
    (3*(491 + 65*57^(1/2)))/(2589 + 343*57^(1/2))}, 
 
   {-4, -3, -2, (8*(-219 + 29*57^(1/2)))/
 
     (-2589 + 343*57^(1/2)), 
 
    (8*(219 + 29*57^(1/2)))/(2589 + 343*57^(1/2))}, 
 
   {0, 0, 1, (-2031 + 269*57^(1/2))/(-2589 + 343*57^(1/2)), 
 
    (2031 + 269*57^(1/2))/(2589 + 343*57^(1/2))}, 
 
   {0, 1, 0, (6*(-385 + 51*57^(1/2)))/
 
     (-2589 + 343*57^(1/2)), 
 
    (6*(385 + 51*57^(1/2)))/(2589 + 343*57^(1/2))}, 
 
   {1, 0, 0, 1, 1}}]
;[o]
               3 (-491 + 65 Sqrt[57])   3 (491 + 65 Sqrt[57])
               ----------------------   ---------------------
3    2    1     -2589 + 343 Sqrt[57]     2589 + 343 Sqrt[57]

               8 (-219 + 29 Sqrt[57])   8 (219 + 29 Sqrt[57])
               ----------------------   ---------------------
-4   -3   -2    -2589 + 343 Sqrt[57]     2589 + 343 Sqrt[57]

               -2031 + 269 Sqrt[57]     2031 + 269 Sqrt[57]
               --------------------     -------------------
0    0    1    -2589 + 343 Sqrt[57]     2589 + 343 Sqrt[57]

               6 (-385 + 51 Sqrt[57])   6 (385 + 51 Sqrt[57])
               ----------------------   ---------------------
0    1    0     -2589 + 343 Sqrt[57]     2589 + 343 Sqrt[57]

1    0    0    1                        1
:[font = input; initialization; preserveAspect]
*)
We will use floating point calculations because of the
radicals which would make the memory fill up.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
P=P//N;
T=Inverse[P].A.P; 
T=Table[If[Abs[T[[i,j]]]>0.0000001,T[[i,j]],0],{i,5},{j,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-2., 0, 0, 0, 0}, {0, -2., 0, 0, 0}, 
 
   {0, 0, -2., 0, 0}, {0, 0, 0, -3.374586088176874243, 0}, 
 
   {0, 0, 0, 0, 34.37458608817687424}}]
;[o]
-2.        0          0          0          0

0          -2.        0          0          0

0          0          -2.        0          0

0          0          0          -3.37459   0

0          0          0          0          34.3746
:[font = input; initialization; preserveAspect]
*)
Exercise 3.5.2
(*
:[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]
*)
ES=Eigensystem[A];
lambda=ES[[1]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0}
;[o]
{0, 0, 0, 0}
:[font = input; initialization; preserveAspect; startGroup]
*)
Q=ES[[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 1}, {0, 0, 0, 0}, {0, 0, 0, 0}, 
 
   {0, 0, 0, 0}}]
;[o]
0   0   0   1

0   0   0   0

0   0   0   0

0   0   0   0
:[font = input; initialization; preserveAspect]
*)
Only one nonzero eigenvector ==> A is not diagonalizable.
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 3.5.3
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[If[i==j,2,1],{i,5},{j,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 1, 1, 1, 1}, {1, 2, 1, 1, 1}, 
 
   {1, 1, 2, 1, 1}, {1, 1, 1, 2, 1}, {1, 1, 1, 1, 2}}]
;[o]
2   1   1   1   1

1   2   1   1   1

1   1   2   1   1

1   1   1   2   1

1   1   1   1   2
:[font = input; initialization; preserveAspect; startGroup]
*)
ES=Eigensystem[A];
lambda=ES[[1]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{1, 1, 1, 1, 6}
;[o]
{1, 1, 1, 1, 6}
:[font = input; initialization; preserveAspect; startGroup]
*)
Q=ES[[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1, 0, 0, 0, 1}, {-1, 0, 0, 1, 0}, 
 
   {-1, 0, 1, 0, 0}, {-1, 1, 0, 0, 0}, {1, 1, 1, 1, 1}}]
;[o]
-1   0    0    0    1

-1   0    0    1    0

-1   0    1    0    0

-1   1    0    0    0

1    1    1    1    1
:[font = input; initialization; preserveAspect]
*)
A is diagonalizable.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Q]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1, -1, -1, -1, 1}, {0, 0, 0, 1, 1}, 
 
   {0, 0, 1, 0, 1}, {0, 1, 0, 0, 1}, {1, 0, 0, 0, 1}}]
;[o]
-1   -1   -1   -1   1

0    0    0    1    1

0    0    1    0    1

0    1    0    0    1

1    0    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}, {0, 1, 0, 0, 0}, 
 
   {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 6}}]
;[o]
1   0   0   0   0

0   1   0   0   0

0   0   1   0   0

0   0   0   1   0

0   0   0   0   6
:[font = input; initialization; preserveAspect]
*)
Exercise 3.5.4
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[i+j-1,{i,5},{j,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2, 3, 4, 5}, {2, 3, 4, 5, 6}, 
 
   {3, 4, 5, 6, 7}, {4, 5, 6, 7, 8}, {5, 6, 7, 8, 9}}]
;[o]
1   2   3   4   5

2   3   4   5   6

3   4   5   6   7

4   5   6   7   8

5   6   7   8   9
:[font = input; initialization; preserveAspect; startGroup]
*)
ES=Eigensystem[A];
lambda=ES[[1]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, (5*(5 - 33^(1/2)))/2, (5*(5 + 33^(1/2)))/2}
;[o]
          5 (5 - Sqrt[33])  5 (5 + Sqrt[33])
{0, 0, 0, ----------------, ----------------}
                 2                 2
:[font = input; initialization; preserveAspect; startGroup]
*)
Q=ES[[2]]//Together; Q//N
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{3., -4., 0, 0, 1.}, {2., -3., 0, 1., 0}, 
 
   {1., -2., 1., 0, 0}, {-1.457427107756338125, 
 
    -0.8430703308172535875, -0.2287135538781690756, 
 
    0.3856432230609154363, 1.}, 
 
   {0.45742710775633811, 0.5930703308172535825, 
 
    0.728713553878169055, 0.8643567769390845275, 1.}}]
;[o]
3.          -4.         0           0           1.

2.          -3.         0           1.          0

1.          -2.         1.          0           0

-1.45743    -0.84307    -0.228714   0.385643    1.

0.457427    0.59307     0.728714    0.864357    1.
:[font = input; initialization; preserveAspect]
*)
A is diagonalizable.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Q]; P//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{3, 2, 1, (-329 + 57*33^(1/2))/
 
     (-717 + 125*33^(1/2)), 
 
    (329 + 57*33^(1/2))/(717 + 125*33^(1/2))}, 
 
   {-4, -3, -2, (2*(-213 + 37*33^(1/2)))/
 
     (-717 + 125*33^(1/2)), 
 
    (2*(213 + 37*33^(1/2)))/(717 + 125*33^(1/2))}, 
 
   {0, 0, 1, (-523 + 91*33^(1/2))/(-717 + 125*33^(1/2)), 
 
    (523 + 91*33^(1/2))/(717 + 125*33^(1/2))}, 
 
   {0, 1, 0, (4*(-155 + 27*33^(1/2)))/(-717 + 125*33^(1/2)), 
 
    (4*(155 + 27*33^(1/2)))/(717 + 125*33^(1/2))}, 
 
   {1, 0, 0, 1, 1}}]
;[o]
               -329 + 57 Sqrt[33]       329 + 57 Sqrt[33]
               -------------------      ------------------
3    2    1    -717 + 125 Sqrt[33]      717 + 125 Sqrt[33]

               2 (-213 + 37 Sqrt[33])   2 (213 + 37 Sqrt[33])
               ----------------------   ---------------------
-4   -3   -2    -717 + 125 Sqrt[33]      717 + 125 Sqrt[33]

               -523 + 91 Sqrt[33]       523 + 91 Sqrt[33]
               -------------------      ------------------
0    0    1    -717 + 125 Sqrt[33]      717 + 125 Sqrt[33]

               4 (-155 + 27 Sqrt[33])   4 (155 + 27 Sqrt[33])
               ----------------------   ---------------------
0    1    0     -717 + 125 Sqrt[33]      717 + 125 Sqrt[33]

1    0    0    1                        1
:[font = input; initialization; preserveAspect; startGroup]
*)
P=P//N;
T=Inverse[P].A.P; 
T=Table[If[Abs[T[[i,j]]]>0.0000001,T[[i,j]],0],{i,5},{j,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0}, {0, 0, 0, -1.86140661634507165, 0}, 
 
   {0, 0, 0, 0, 26.86140661634507165}}]
;[o]
0          0          0          0          0

0          0          0          0          0

0          0          0          0          0

0          0          0          -1.86141   0

0          0          0          0          26.8614
:[font = input; initialization; preserveAspect]
*)
Exercise 3.5.5
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[i+j,{i,5},{j,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 3, 4, 5, 6}, {3, 4, 5, 6, 7}, 
 
   {4, 5, 6, 7, 8}, {5, 6, 7, 8, 9}, {6, 7, 8, 9, 10}}]
;[o]
2    3    4    5    6

3    4    5    6    7

4    5    6    7    8

5    6    7    8    9

6    7    8    9    10
:[font = input; initialization; preserveAspect; startGroup]
*)
ES=Eigensystem[A];
lambda=ES[[1]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 5*(3 - 11^(1/2)), 5*(3 + 11^(1/2))}
;[o]
{0, 0, 0, 5 (3 - Sqrt[11]), 5 (3 + Sqrt[11])}
:[font = input; initialization; preserveAspect; startGroup]
*)
Q=ES[[2]]//Together; Q//N
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{3., -4., 0, 0, 1.}, {2., -3., 0, 1., 0}, 
 
   {1., -2., 1., 0, 0}, {-1.376178511530114289, 
 
    -0.782133883647585752, -0.1880892557650572153, 
 
    0.4059553721174714278, 1.}, 
 
   {0.5190356543872570998, 0.6392767407904428248, 
 
    0.7595178271936285498, 0.879758913596814275, 1.}}]
;[o]
3.          -4.         0           0           1.

2.          -3.         0           1.          0

1.          -2.         1.          0           0

-1.37618    -0.782134   -0.188089   0.405955    1.

0.519036    0.639277    0.759518    0.879759    1.
:[font = input; initialization; preserveAspect]
*)
A is diagonalizable.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Q]; P//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{3, 2, 1, (-93 + 28*11^(1/2))/
 
     (-179 + 54*11^(1/2)), 
 
    (93 + 28*11^(1/2))/(179 + 54*11^(1/2))}, 
 
   {-4, -3, -2, (-229 + 69*11^(1/2))/
 
     (2*(-179 + 54*11^(1/2))), 
 
    (229 + 69*11^(1/2))/(2*(179 + 54*11^(1/2)))}, 
 
   {0, 0, 1, (-136 + 41*11^(1/2))/(-179 + 54*11^(1/2)), 
 
    (136 + 41*11^(1/2))/(179 + 54*11^(1/2))}, 
 
   {0, 1, 0, (5*(-63 + 19*11^(1/2)))/
 
     (2*(-179 + 54*11^(1/2))), 
 
    (5*(63 + 19*11^(1/2)))/(2*(179 + 54*11^(1/2)))}, 
 
   {1, 0, 0, 1, 1}}]
;[o]
               -93 + 28 Sqrt[11]        93 + 28 Sqrt[11]
               ------------------       -----------------
3    2    1    -179 + 54 Sqrt[11]       179 + 54 Sqrt[11]

                 -229 + 69 Sqrt[11]       229 + 69 Sqrt[11]
               ----------------------   ---------------------
-4   -3   -2   2 (-179 + 54 Sqrt[11])   2 (179 + 54 Sqrt[11])

               -136 + 41 Sqrt[11]       136 + 41 Sqrt[11]
               ------------------       -----------------
0    0    1    -179 + 54 Sqrt[11]       179 + 54 Sqrt[11]

               5 (-63 + 19 Sqrt[11])    5 (63 + 19 Sqrt[11])
               ----------------------   ---------------------
0    1    0    2 (-179 + 54 Sqrt[11])   2 (179 + 54 Sqrt[11])

1    0    0    1                        1
:[font = input; initialization; preserveAspect; startGroup]
*)
P=P//N;
T=Inverse[P].A.P; 
T=Table[If[Abs[T[[i,j]]]>0.0000001,T[[i,j]],0],{i,5},{j,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0}, {0, 0, 0, -1.583123951776999246, 0}, 
 
   {0, 0, 0, 0, 31.58312395177699924}}]
;[o]
0          0          0          0          0

0          0          0          0          0

0          0          0          0          0

0          0          0          -1.58312   0

0          0          0          0          31.5831
:[font = input; initialization; preserveAspect]
*)
Exercise 3.5.6
(*
:[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]
*)
ES=Eigensystem[A];
ES[[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 1}, {0, 0, 0, 0}, {0, 0, 0, 0}, 
 
   {0, 0, 0, 0}}]
;[o]
0   0   0   1

0   0   0   0

0   0   0   0

0   0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
M=NullSpace[ES[[2]]]
(*
:[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[{ES[[2,1]]},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 3.5.7
(*
:[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]
*)
ES=Eigensystem[A];
ES[[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-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]
*)
M=NullSpace[ES[[2]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{}
;[o]
{}
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[ES[[2]]]
(*
:[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 3.5.8
(*
:[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]
*)
ES=Eigensystem[A];
ES[[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-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]
*)
M=NullSpace[ES[[2]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{}
;[o]
{}
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[ES[[2]]]
(*
:[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 3.5.9
(*
:[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]
*)
ES=Eigensystem[A];
ES[[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-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]
*)
M=NullSpace[ES[[2]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{}
;[o]
{}
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[ES[[2]]]
(*
:[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
^*)