(*^
::[	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"; 
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	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"; 
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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[#],#]&
<<LinearAlgebra`Master`
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 3.3.1
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M=Array[Max,{3,3}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2, 3}, {2, 2, 3}, {3, 3, 3}}]
;[o]
1   2   3

2   2   3

3   3   3
:[font = input; initialization; preserveAspect; startGroup]
*)
Id[3]=IdentityMatrix[3];
p=Det[x Id[3]-M]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
-3 - 11*x - 6*x^2 + x^3
;[o]
               2    3
-3 - 11 x - 6 x  + x
:[font = input; initialization; preserveAspect; startGroup]
*)
lambda=Solve[p==0,x][[1,1,2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
2 + (23*2^(1/3))/(1107 + 3*I*9843^(1/2))^(1/3) + 
 
  (1107 + 3*I*9843^(1/2))^(1/3)/(3*2^(1/3))
;[o]
                 1/3                                    1/3
             23 2                (1107 + 3 I Sqrt[9843])
2 + -------------------------- + --------------------------
                           1/3                1/3
    (1107 + 3 I Sqrt[9843])                3 2
:[font = input; initialization; preserveAspect; startGroup]
*)
q=p/.{x->r};
CM=r Id[3]-M
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1 + r, -2, -3}, {-2, -2 + r, -3}, 
 
   {-3, -3, -3 + r}}]
;[o]
-1 + r   -2       -3

-2       -2 + r   -3

-3       -3       -3 + r
:[font = input; initialization; preserveAspect; startGroup]
*)
CM[[1]]=CM[[1]]/CM[[1,1]]; CM
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, -2/(-1 + r), -3/(-1 + r)}, {-2, -2 + r, -3}, 
 
   {-3, -3, -3 + r}}]
;[o]
           -2       -3
         ------   ------
1        -1 + r   -1 + r



-2       -2 + r   -3



-3       -3       -3 + r
:[font = input; initialization; preserveAspect; startGroup]
*)
CM[[2]]=CM[[2]]+2CM[[1]];
CM[[3]]=CM[[3]]+3CM[[1]];
CM=CM//Together
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, -2/(-1 + r), -3/(-1 + r)}, 
 
   {0, (-2 - 3*r + r^2)/(-1 + r), (-3*(1 + r))/(-1 + r)}, 
 
   {0, (-3*(1 + r))/(-1 + r), (-6 - 4*r + r^2)/(-1 + r)}}]
;[o]

                  -2              -3
                ------          ------
1               -1 + r          -1 + r

                            2
                -2 - 3 r + r    -3 (1 + r)
                -------------   ----------
0                  -1 + r         -1 + r

                                            2
                -3 (1 + r)      -6 - 4 r + r
                ----------      -------------
0                 -1 + r           -1 + r
:[font = input; initialization; preserveAspect; startGroup]
*)
CM[[2]]=CM[[2]]/CM[[2,2]];
CM=CM//Together
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, -2/(-1 + r), -3/(-1 + r)}, 
 
   {0, 1, (-3*(1 + r))/(-2 - 3*r + r^2)}, 
 
   {0, (-3*(1 + r))/(-1 + r), (-6 - 4*r + r^2)/(-1 + r)}}]
;[o]

                  -2              -3
                ------          ------
1               -1 + r          -1 + r

                                 -3 (1 + r)
                                -------------
                                            2
0               1               -2 - 3 r + r

                                            2
                -3 (1 + r)      -6 - 4 r + r
                ----------      -------------
0                 -1 + r           -1 + r
:[font = input; initialization; preserveAspect; startGroup]
*)
CM[[3]]=CM[[3]]-CM[[3,2]]CM[[2]];
CM[[1]]=CM[[1]]-CM[[1,2]]CM[[2]];
CM=CM//Together; CM//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, (-3*r)/(-2 - 3*r + r^2)}, 
 
   {0, 1, (-3*(1 + r))/(-2 - 3*r + r^2)}, 
 
   {0, 0, (-3 - 11*r - 6*r^2 + r^3)/(-2 - 3*r + r^2)}}]
;[o]
            -3 r
        -------------
                    2
1   0   -2 - 3 r + r

         -3 (1 + r)
        -------------
                    2
0   1   -2 - 3 r + r

                       2    3
        -3 - 11 r - 6 r  + r
        ---------------------
                        2
0   0       -2 - 3 r + r
:[font = input; initialization; preserveAspect; startGroup]
*)
CM[[3,3]]=0; CM
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, (-3*r)/(-2 - 3*r + r^2)}, 
 
   {0, 1, (-3*(1 + r))/(-2 - 3*r + r^2)}, {0, 0, 0}}]
;[o]
                                    -3 r
                                -------------
                                            2
1               0               -2 - 3 r + r

                                 -3 (1 + r)
                                -------------
                                            2
0               1               -2 - 3 r + r




0               0               0
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[CM]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(3*r)/(-2 - 3*r + r^2), 
 
    (3*(1 + r))/(-2 - 3*r + r^2), 1}}]
;[o]
     3 r          3 (1 + r)
-------------   -------------
            2               2
-2 - 3 r + r    -2 - 3 r + r    1
:[font = input; initialization; preserveAspect]
*)
lambda=Solve[p==0,x];
(*
:[font = input; initialization; preserveAspect]
*)
Do[r[i]=lambda[[i,1,2]],{i,1,3}]
(*
:[font = input; initialization; preserveAspect]
*)
Do[ES[i]=%%%/.{r->r[i]},{i,1,3}]
(*
:[font = input; initialization; preserveAspect]
*)
P=Transpose[Join[ES[1],ES[2],ES[3]]];
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[r Id[3]-M,ZeroTest->(PolynomialMod[#,q]==0&)] \
         //Together
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(-3 - 5*r + r^2)/(3*r), (3 + 2*r)/(3*r), 1}}]
;[o]
            2
-3 - 5 r + r    3 + 2 r
-------------   -------
     3 r          3 r           1
:[font = input; initialization; preserveAspect]
*)
In order not to repeat parts of the exercise, the equivalence
of the above result with NullSpace[CM] will be shown at the
end of the next exercise.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
NS=%
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(-3 - 5*r + r^2)/(3*r), (3 + 2*r)/(3*r), 1}}]
;[o]
            2
-3 - 5 r + r    3 + 2 r
-------------   -------
     3 r          3 r           1
:[font = input; initialization; preserveAspect]
*)
Exercise 3.3.2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
P=N[P];
Unprotect[D];
D=Inverse[P].M.P;
D1[n_]:=Table[If[Abs[Re[D[[i,j]]]]<=0.000001,                 \
                 I Im[D[[i,j]]],D[[i,j]]],{i,n},{j,n}]
Diag[n_]:=Table[If[Abs[Im[D1[n][[i,j]]]]<=0.000001,           \
                   Re[D1[n][[i,j]]],D1[n][[i,j]]],{i,n},{j,n}]
Diag[3]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{7.516538485191787522, 0, 0}, 
 
   {0, -1.177616778714116014, 0}, 
 
   {0, 0, -0.3389217064776715073}}]
;[o]
7.51654     0           0

0           -1.17762    0

0           0           -0.338922
:[font = input; initialization; preserveAspect]
*)
Equivalence from previous exercise:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[ES[i]=NS/.{r->r[i]},{i,1,3}]
P=Transpose[Join[ES[1],ES[2],ES[3]]];
P=N[P];
D=Inverse[P].M.P;
Diag[3]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{7.516538485191787521, 0, 0}, 
 
   {0, -1.177616778714116014, 0}, 
 
   {0, 0, -0.3389217064776715073}}]
;[o]
7.51654     0           0

0           -1.17762    0

0           0           -0.338922
:[font = input; initialization; preserveAspect]
*)
Exercise 3.3.3
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M=Array[Max,{4,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2, 3, 4}, {2, 2, 3, 4}, {3, 3, 3, 4}, 
 
   {4, 4, 4, 4}}]
;[o]
1   2   3   4

2   2   3   4

3   3   3   4

4   4   4   4
:[font = input; initialization; preserveAspect; startGroup]
*)
x IdentityMatrix[4]-M
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1 + x, -2, -3, -4}, {-2, -2 + x, -3, -4}, 
 
   {-3, -3, -3 + x, -4}, {-4, -4, -4, -4 + x}}]
;[o]
-1 + x   -2       -3       -4

-2       -2 + x   -3       -4

-3       -3       -3 + x   -4

-4       -4       -4       -4 + x
:[font = input; initialization; preserveAspect; startGroup]
*)
CM=Join[RowReduce[Array[%[[#]]&,3]],{{0,0,0,0}}];
CM=CM//Together; CM//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0, (-4*x^2)/(-3 - 11*x - 6*x^2 + x^3)}, 
 
   {0, 1, 0, (-4*(x + x^2))/(-3 - 11*x - 6*x^2 + x^3)}, 
 
   {0, 0, 1, (-4*(1 + 3*x + x^2))/
 
     (-3 - 11*x - 6*x^2 + x^3)}, {0, 0, 0, 0}}]
;[o]
                        2
                    -4 x
            ---------------------
                           2    3
1   0   0   -3 - 11 x - 6 x  + x

                          2
                 -4 (x + x )
            ---------------------
                           2    3
0   1   0   -3 - 11 x - 6 x  + x

                             2
              -4 (1 + 3 x + x )
            ---------------------
                           2    3
0   0   1   -3 - 11 x - 6 x  + x

0   0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[r];
p=Det[r IdentityMatrix[4]-M]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
-4 - 23*r - 35*r^2 - 10*r^3 + r^4
;[o]
                2       3    4
-4 - 23 r - 35 r  - 10 r  + r
:[font = input; initialization; preserveAspect; startGroup]
*)
Solve[p==0,r];
r=r/.%;
P=Transpose[Table[NullSpace[CM][[1]]/.{x->r[[i]]//N},{i,4}]//N]//Re
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1.174018626437017946, 1.132327326223258363, 
 
    -1.126230617252018513, 0.6679219174657781049}, 
 
   {-0.6021957184026057897, -1.067892654641997108, 
 
    2.700236934218879539, 0.7198514388257233394}, 
 
   {0.262935403955374566, -1.193095366428914725, 
 
    -2.647587929910311594, 0.8277478923838517618}, 
 
   {1., 1., 1., 1.}}]
;[o]
-1.17402    1.13233     -1.12623    0.667922

-0.602196   -1.06789    2.70024     0.719851

0.262935    -1.1931     -2.64759    0.827748

1.          1.          1.          1.
:[font = input; initialization; preserveAspect]
*)
Although it is not specified in the exercises, we will use
floating point representations in this chapter, since exact
calculations here would overflow the capacity of personal
computers at the present time. 
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
D=Inverse[P].M.P;
Diag[4]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-2.053115763536996677, 0, 0, 0}, 
 
   {0, -0.5146427793906138797, 0, 0}, 
 
   {0, 0, -0.2943264517738022657, 0}, 
 
   {0, 0, 0, 12.86208499470141282}}]
;[o]
-2.05312    0           0           0

0           -0.514643   0           0

0           0           -0.294326   0

0           0           0           12.8621
:[font = input; initialization; preserveAspect]
*)
Exercise 3.3.4
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M=Array[Min,{4,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 1, 1, 1}, {1, 2, 2, 2}, {1, 2, 3, 3}, 
 
   {1, 2, 3, 4}}]
;[o]
1   1   1   1

1   2   2   2

1   2   3   3

1   2   3   4
:[font = input; initialization; preserveAspect; startGroup]
*)
x IdentityMatrix[4]-M
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1 + x, -1, -1, -1}, {-1, -2 + x, -2, -2}, 
 
   {-1, -2, -3 + x, -3}, {-1, -2, -3, -4 + x}}]
;[o]
-1 + x   -1       -1       -1

-1       -2 + x   -2       -2

-1       -2       -3 + x   -3

-1       -2       -3       -4 + x
:[font = input; initialization; preserveAspect; startGroup]
*)
CM=Join[RowReduce[Array[%[[#]]&,3]],{{0,0,0,0}}];
CM=CM//Together; CM//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0, -(x^2/(-1 + 5*x - 6*x^2 + x^3))}, 
 
   {0, 1, 0, (x - 2*x^2)/(-1 + 5*x - 6*x^2 + x^3)}, 
 
   {0, 0, 1, (-1 + 4*x - 3*x^2)/(-1 + 5*x - 6*x^2 + x^3)}, 
 
   {0, 0, 0, 0}}]
;[o]
                        2
                       x
            -(--------------------)
                            2    3
1   0   0     -1 + 5 x - 6 x  + x

                         2
                  x - 2 x
            --------------------
                          2    3
0   1   0   -1 + 5 x - 6 x  + x

                            2
              -1 + 4 x - 3 x
            --------------------
                          2    3
0   0   1   -1 + 5 x - 6 x  + x

0   0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[r];
p=Det[r IdentityMatrix[4]-M]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
1 - 7*r + 15*r^2 - 10*r^3 + r^4
;[o]
              2       3    4
1 - 7 r + 15 r  - 10 r  + r
:[font = input; initialization; preserveAspect; startGroup]
*)
Solve[p==0,r];
r=r/.%;
P=Transpose[Table[NullSpace[CM][[1]]/.{x->r[[i]]//N},{i,4}]//N]//Re
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1., 0.3472963553338606977, 
 
    -1.879385241571816773, 1.53208888623795607}, 
 
   {-1., 0.6527036446661393023, 2.879385241571816774, 
 
    -0.5320888862379560712}, 
 
   {0., 0.8793852415718167681, -2.532088886237956071, 
 
    -1.347296355333860697}, {1., 1., 1., 1.}}]
;[o]
-1.         0.347296    -1.87939    1.53209

-1.         0.652704    2.87939     -0.532089

0.          0.879385    -2.53209    -1.3473

1.          1.          1.          1.
:[font = input; initialization; preserveAspect; startGroup]
*)
Inverse[P].M.P;
Diag[4]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-2.053115763536996677, 0, 0, 0}, 
 
   {0, -0.5146427793906138797, 0, 0}, 
 
   {0, 0, -0.2943264517738022657, 0}, 
 
   {0, 0, 0, 12.86208499470141282}}]
;[o]
-2.05312    0           0           0

0           -0.514643   0           0

0           0           -0.294326   0

0           0           0           12.8621
:[font = input; initialization; preserveAspect]
*)
Exercise 3.3.5
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M=Array[Max,{5,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2, 3, 4, 5}, {2, 2, 3, 4, 5}, 
 
   {3, 3, 3, 4, 5}, {4, 4, 4, 4, 5}, {5, 5, 5, 5, 5}}]
;[o]
1   2   3   4   5

2   2   3   4   5

3   3   3   4   5

4   4   4   4   5

5   5   5   5   5
:[font = input; initialization; preserveAspect; startGroup]
*)
x IdentityMatrix[5]-M
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1 + x, -2, -3, -4, -5}, 
 
   {-2, -2 + x, -3, -4, -5}, {-3, -3, -3 + x, -4, -5}, 
 
   {-4, -4, -4, -4 + x, -5}, {-5, -5, -5, -5, -5 + x}}]
;[o]
-1 + x   -2       -3       -4       -5

-2       -2 + x   -3       -4       -5

-3       -3       -3 + x   -4       -5

-4       -4       -4       -4 + x   -5

-5       -5       -5       -5       -5 + x
:[font = input; initialization; preserveAspect; startGroup]
*)
CM=Join[RowReduce[Array[%[[#]]&,4]],{{0,0,0,0,0}}];
CM=CM//Together; CM//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0, 0, (-5*x^3)/
 
     (-4 - 23*x - 35*x^2 - 10*x^3 + x^4)}, 
 
   {0, 1, 0, 0, (-5*(x^2 + x^3))/
 
     (-4 - 23*x - 35*x^2 - 10*x^3 + x^4)}, 
 
   {0, 0, 1, 0, (-5*(x + 3*x^2 + x^3))/
 
     (-4 - 23*x - 35*x^2 - 10*x^3 + x^4)}, 
 
   {0, 0, 0, 1, (-5*(1 + 5*x + 6*x^2 + x^3))/
 
     (-4 - 23*x - 35*x^2 - 10*x^3 + x^4)}, {0, 0, 0, 0, 0}}]
;[o]
                                3
                            -5 x
                ------------------------------
                                2       3    4
1   0   0   0   -4 - 23 x - 35 x  - 10 x  + x

                              2    3
                         -5 (x  + x )
                ------------------------------
                                2       3    4
0   1   0   0   -4 - 23 x - 35 x  - 10 x  + x

                                 2    3
                      -5 (x + 3 x  + x )
                ------------------------------
                                2       3    4
0   0   1   0   -4 - 23 x - 35 x  - 10 x  + x

                                    2    3
                   -5 (1 + 5 x + 6 x  + x )
                ------------------------------
                                2       3    4
0   0   0   1   -4 - 23 x - 35 x  - 10 x  + x

0   0   0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[r]
p=Det[r IdentityMatrix[5]-M]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
-5 - 39*r - 98*r^2 - 85*r^3 - 15*r^4 + r^5
;[o]
                2       3       4    5
-5 - 39 r - 98 r  - 85 r  - 15 r  + r
:[font = input; initialization; preserveAspect; startGroup]
*)
NSolve[p==0,r];
r=r/.%;
P=Transpose[Table[NullSpace[CM][[1]]/.{x->r[[i]]//N},{i,5}]//N]//Re;
P//Together//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{-1.152130131983863609, 1.109051588148310365, 
 
    -1.102464924693090638, 1.100489567734286696, 
 
    0.6450539007943571859}, 
 
   {-0.7904047325882884809, -0.3699093956738938238, 
 
    1.75571471841964394, -2.873369373142965157, 
 
    0.6779687829855035194}, 
 
   {-0.1805220637640063452, -1.355582628414694461, 
 
    0.06214132621707466106, 3.528485285351223587, 
 
    0.7454780806104025686}, 
 
   {0.4860377232103835421, -0.5335366899311692582, 
 
    -1.792535670836337992, -2.810991923402531029, 
 
    0.8510265609596547179}, {1., 1., 1., 1., 1.}}]
;[o]
-1.15213    1.10905     -1.10246    1.10049    0.645054

-0.790405   -0.369909   1.75571     -2.87337   0.677969

-0.180522   -1.35558    0.0621413   3.52849    0.745478

0.486038    -0.533537   -1.79254    -2.81099   0.851027

1.          1.          1.          1.         1.
:[font = input; initialization; preserveAspect; startGroup]
*)
D=Inverse[P].M.P;
Diag[5]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-3.185096025628874463, 0, 0, 0, 0}, 
 
   {0, -0.749885629357235896, 0, 0, 0}, 
 
   {0, 0, -0.3857227544635501227, 0, 0}, 
 
   {0, 0, 0, -0.2769322172999294724, 0}, 
 
   {0, 0, 0, 0, 19.59763662674958996}}]
;[o]
-3.1851     0           0           0           0

0           -0.749886   0           0           0

0           0           -0.385723   0           0

0           0           0           -0.276932   0

0           0           0           0           19.5976
:[font = input; initialization; preserveAspect]
*)
Exercise 3.3.6
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M=Table[If[i>3,If[j>3,Min[i,j],0],If[j>3,0,Max[i,j]]], \
                                           {i,6},{j,6}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2, 3, 0, 0, 0}, {2, 2, 3, 0, 0, 0}, 
 
   {3, 3, 3, 0, 0, 0}, {0, 0, 0, 4, 4, 4}, 
 
   {0, 0, 0, 4, 5, 5}, {0, 0, 0, 4, 5, 6}}]
;[o]
1   2   3   0   0   0

2   2   3   0   0   0

3   3   3   0   0   0

0   0   0   4   4   4

0   0   0   4   5   5

0   0   0   4   5   6
:[font = input; initialization; preserveAspect]
*)
Trying to solve for the 6*6 matrix as a whole, we fail. 
However, we notice that by finding diagonalizing matrices 
for the upper left and the lower right corner of M we can 
then construct the diagonalizing matrix of M by the same 
pattern M was constructed.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M1=SubMatrix[M,{1,1},{3,3}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2, 3}, {2, 2, 3}, {3, 3, 3}}]
;[o]
1   2   3

2   2   3

3   3   3
:[font = input; initialization; preserveAspect; startGroup]
*)
M2=SubMatrix[M,{4,4},{3,3}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{4, 4, 4}, {4, 5, 5}, {4, 5, 6}}]
;[o]
4   4   4

4   5   5

4   5   6
:[font = input; initialization; preserveAspect; startGroup]
*)
x Id[3]-M1
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1 + x, -2, -3}, {-2, -2 + x, -3}, 
 
   {-3, -3, -3 + x}}]
;[o]
-1 + x   -2       -3

-2       -2 + x   -3

-3       -3       -3 + x
:[font = input; initialization; preserveAspect; startGroup]
*)
CM=Join[RowReduce[Array[%[[#]]&,2]],{{0,0,0}}];
CM=CM//Together; CM//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, (-3*x)/(-2 - 3*x + x^2)}, 
 
   {0, 1, (-3*(1 + x))/(-2 - 3*x + x^2)}, {0, 0, 0}}]
;[o]
            -3 x
        -------------
                    2
1   0   -2 - 3 x + x

         -3 (1 + x)
        -------------
                    2
0   1   -2 - 3 x + x

0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[r]
p=Det[r Id[3]-M1]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
-3 - 11*r - 6*r^2 + r^3
;[o]
               2    3
-3 - 11 r - 6 r  + r
:[font = input; initialization; preserveAspect; startGroup]
*)
Solve[p==0,r];
r=r/.%
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{2 + (23*2^(1/3))/(1107 + 3*I*9843^(1/2))^(1/3) + 
 
   (1107 + 3*I*9843^(1/2))^(1/3)/(3*2^(1/3)), 
 
  2 - (23*(1 + I*3^(1/2)))/
 
    (2^(2/3)*(1107 + 3*I*9843^(1/2))^(1/3)) - 
 
   ((1 - I*3^(1/2))*(1107 + 3*I*9843^(1/2))^(1/3))/
 
    (6*2^(1/3)), 2 - (23*(1 - I*3^(1/2)))/
 
    (2^(2/3)*(1107 + 3*I*9843^(1/2))^(1/3)) - 
 
   ((1 + I*3^(1/2))*(1107 + 3*I*9843^(1/2))^(1/3))/
 
    (6*2^(1/3))}
;[o]
                  1/3
              23 2
{2 + -------------------------- + 
                            1/3
     (1107 + 3 I Sqrt[9843])
 
                          1/3
   (1107 + 3 I Sqrt[9843])
   --------------------------, 
                1/3
             3 2
 
            23 (1 + I Sqrt[3])
  2 - ------------------------------- - 
       2/3                        1/3
      2    (1107 + 3 I Sqrt[9843])
 
                                          1/3
   (1 - I Sqrt[3]) (1107 + 3 I Sqrt[9843])
   ------------------------------------------, 
                        1/3
                     6 2
 
            23 (1 - I Sqrt[3])
  2 - ------------------------------- - 
       2/3                        1/3
      2    (1107 + 3 I Sqrt[9843])
 
                                          1/3
   (1 + I Sqrt[3]) (1107 + 3 I Sqrt[9843])
   ------------------------------------------}
                        1/3
                     6 2
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Table[NullSpace[CM][[1]]/.{x->r[[i]]//N},{i,3}]//N]//Re
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0.7058061989910869513, -1.210032909502607794, 
 
    1.170893377178187511}, 
 
   {0.7997066294061755558, -0.1825060167354308772, 
 
    -2.283867279337411347}, {1., 1., 1.}}]
;[o]
0.705806    -1.21003    1.17089

0.799707    -0.182506   -2.28387

1.          1.          1.
:[font = input; initialization; preserveAspect; startGroup]
*)
D=Inverse[P].M1.P;
Diag[3]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{7.516538485191787522, 0, 0}, 
 
   {0, -1.177616778714116014, 0}, 
 
   {0, 0, -0.3389217064776715072}}]
;[o]
7.51654     0           0

0           -1.17762    0

0           0           -0.338922
:[font = input; initialization; preserveAspect; startGroup]
*)
P1=P
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0.7058061989910869513, -1.210032909502607794, 
 
    1.170893377178187511}, 
 
   {0.7997066294061755558, -0.1825060167354308772, 
 
    -2.283867279337411347}, {1., 1., 1.}}]
;[o]
0.705806    -1.21003    1.17089

0.799707    -0.182506   -2.28387

1.          1.          1.
:[font = input; initialization; preserveAspect; startGroup]
*)
x Id[3]-M2
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-4 + x, -4, -4}, {-4, -5 + x, -5}, 
 
   {-4, -5, -6 + x}}]
;[o]
-4 + x   -4       -4

-4       -5 + x   -5

-4       -5       -6 + x
:[font = input; initialization; preserveAspect; startGroup]
*)
CM=Join[RowReduce[Array[%[[#]]&,2]],{{0,0,0}}];
CM=CM//Together; CM//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, (-4*x)/(4 - 9*x + x^2)}, 
 
   {0, 1, (4 - 5*x)/(4 - 9*x + x^2)}, {0, 0, 0}}]
;[o]
            -4 x
        ------------
                   2
1   0   4 - 9 x + x

          4 - 5 x
        ------------
                   2
0   1   4 - 9 x + x

0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[r]
p=Det[r Id[3]-M2]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
-4 + 17*r - 15*r^2 + r^3
;[o]
                2    3
-4 + 17 r - 15 r  + r
:[font = input; initialization; preserveAspect; startGroup]
*)
Solve[p==0,r];
r=r/.%
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{5 + (58*2^(1/3))/(4563 + 3*I*27903^(1/2))^(1/3) + 
 
   (4563 + 3*I*27903^(1/2))^(1/3)/(3*2^(1/3)), 
 
  5 - (29*2^(1/3)*(1 + I*3^(1/2)))/
 
    (4563 + 3*I*27903^(1/2))^(1/3) - 
 
   ((1 - I*3^(1/2))*(4563 + 3*I*27903^(1/2))^(1/3))/
 
    (6*2^(1/3)), 5 - (29*2^(1/3)*(1 - I*3^(1/2)))/
 
    (4563 + 3*I*27903^(1/2))^(1/3) - 
 
   ((1 + I*3^(1/2))*(4563 + 3*I*27903^(1/2))^(1/3))/
 
    (6*2^(1/3))}
;[o]
                   1/3
               58 2
{5 + --------------------------- + 
                             1/3
     (4563 + 3 I Sqrt[27903])
 
                           1/3
   (4563 + 3 I Sqrt[27903])
   ---------------------------, 
                1/3
             3 2
 
            1/3
        29 2    (1 + I Sqrt[3])
  5 - --------------------------- - 
                              1/3
      (4563 + 3 I Sqrt[27903])
 
                                           1/3
   (1 - I Sqrt[3]) (4563 + 3 I Sqrt[27903])
   -------------------------------------------, 
                        1/3
                     6 2
 
            1/3
        29 2    (1 - I Sqrt[3])
  5 - --------------------------- - 
                              1/3
      (4563 + 3 I Sqrt[27903])
 
                                           1/3
   (1 + I Sqrt[3]) (4563 + 3 I Sqrt[27903])
   -------------------------------------------}
                        1/3
                     6 2
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Table[NullSpace[CM][[1]]/.{x->r[[i]]//N},{i,3}]//N]//Re
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0.7876810238413977099, 1.139220666558505532, 
 
    -1.114401690399903244}, 
 
   {0.9274736508396215261, -2.045710408374901746, 
 
    -0.1317632424647197777}, {1., 1., 1.}}]
;[o]
0.787681    1.13922     -1.1144

0.927474    -2.04571    -0.131763

1.          1.          1.
:[font = input; initialization; preserveAspect; startGroup]
*)
D=Inverse[P].M2.P;
Diag[3]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{13.78809234956369847, 0, 0}, 
 
   {0, 0.328330624359513393, 0}, 
 
   {0, 0, 0.8835770260767881354}}]
;[o]
13.7881    0          0

0          0.328331   0

0          0          0.883577
:[font = input; initialization; preserveAspect; startGroup]
*)
P2=P
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0.7876810238413977099, 1.139220666558505532, 
 
    -1.114401690399903244}, 
 
   {0.9274736508396215261, -2.045710408374901746, 
 
    -0.1317632424647197777}, {1., 1., 1.}}]
;[o]
0.787681    1.13922     -1.1144

0.927474    -2.04571    -0.131763

1.          1.          1.
:[font = input; initialization; preserveAspect; startGroup]
*)
ZeroMatrix=Array[0&,{3,3}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}]
;[o]
0   0   0

0   0   0

0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
P=BlockMatrix[{{P1,ZeroMatrix},{ZeroMatrix,P2}}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0.7058061989910869513, -1.210032909502607794, 
 
    1.170893377178187511, 0, 0, 0}, 
 
   {0.7997066294061755558, -0.1825060167354308772, 
 
    -2.283867279337411347, 0, 0, 0}, {1., 1., 1., 0, 0, 0}, 
 
   {0, 0, 0, 0.7876810238413977099, 1.139220666558505532, 
 
    -1.114401690399903244}, 
 
   {0, 0, 0, 0.9274736508396215261, -2.045710408374901746, 
 
    -0.1317632424647197777}, {0, 0, 0, 1., 1., 1.}}]
;[o]
0.705806    -1.21003    1.17089     0           0
 
      0

0.799707    -0.182506   -2.28387    0           0
 
      0

1.          1.          1.          0           0
 
      0

0           0           0           0.787681    1.13922
 
      -1.1144

0           0           0           0.927474    -2.04571
 
      -0.131763

0           0           0           1.          1.
 
      1.
:[font = input; initialization; preserveAspect; startGroup]
*)
D=Inverse[P].M.P;
Diag[6]//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{7.516538485191787522, 0, 0, 0, 0, 0}, 
 
   {0, -1.177616778714116014, 0, 0, 0, 0}, 
 
   {0, 0, -0.3389217064776715072, 0, 0, 0}, 
 
   {0, 0, 0, 13.78809234956369847, 0, 0}, 
 
   {0, 0, 0, 0, 0.328330624359513393, 0}, 
 
   {0, 0, 0, 0, 0, 0.8835770260767881354}}]
;[o]
7.51654   0          0           0         0
 
   0

0         -1.17762   0           0         0
 
   0

0         0          -0.338922   0         0
 
   0

0         0          0           13.7881   0
 
   0

0         0          0           0         0.328331
 
   0

0         0          0           0         0          0.8835\
 
   77
^*)