(*^
::[	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.";

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	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"; 
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	fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7,  12, "Times"; 
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	fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
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	paletteColors = 128; automaticGrouping; currentKernel; 
]
:[font = input; initialization; preserveAspect]
*)
$Post:=If[MatrixQ[#],MatrixForm[#],#]&
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 3.2.1
(*
:[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]
*)
lambda=x/.Solve[Det[x*IdentityMatrix[5]-A]==0,x]
(*
:[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]
*)
ES[1]=NullSpace[lambda[[1]]IdentityMatrix[5]-A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{3, -4, 0, 0, 1}, {2, -3, 0, 1, 0}, 
 
   {1, -2, 1, 0, 0}}]
;[o]
3    -4   0    0    1

2    -3   0    1    0

1    -2   1    0    0
:[font = input; initialization; preserveAspect; startGroup]
*)
ES[2]=Together[NullSpace[lambda[[4]]IdentityMatrix[5]-A]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(3*(-491 + 65*57^(1/2)))/
 
     (-2589 + 343*57^(1/2)), 
 
    (8*(-219 + 29*57^(1/2)))/(-2589 + 343*57^(1/2)), 
 
    (-2031 + 269*57^(1/2))/(-2589 + 343*57^(1/2)), 
 
    (6*(-385 + 51*57^(1/2)))/(-2589 + 343*57^(1/2)), 1}}]
;[o]
3 (-491 + 65 Sqrt[57])   8 (-219 + 29 Sqrt[57])
----------------------   ----------------------
 -2589 + 343 Sqrt[57]     -2589 + 343 Sqrt[57]
 
   -2031 + 269 Sqrt[57]     6 (-385 + 51 Sqrt[57])
   --------------------     ----------------------
   -2589 + 343 Sqrt[57]      -2589 + 343 Sqrt[57]
 


   1
:[font = input; initialization; preserveAspect; startGroup]
*)
ES[3]=Together[NullSpace[lambda[[5]]IdentityMatrix[5]-A]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(3*(491 + 65*57^(1/2)))/(2589 + 343*57^(1/2)), 
 
    (8*(219 + 29*57^(1/2)))/(2589 + 343*57^(1/2)), 
 
    (2031 + 269*57^(1/2))/(2589 + 343*57^(1/2)), 
 
    (6*(385 + 51*57^(1/2)))/(2589 + 343*57^(1/2)), 1}}]
;[o]
3 (491 + 65 Sqrt[57])   8 (219 + 29 Sqrt[57])
---------------------   ---------------------
 2589 + 343 Sqrt[57]     2589 + 343 Sqrt[57]
 
   2031 + 269 Sqrt[57]     6 (385 + 51 Sqrt[57])
   -------------------     ---------------------
   2589 + 343 Sqrt[57]      2589 + 343 Sqrt[57]
 


   1
:[font = input; initialization; preserveAspect]
*)
With Mathematica rendering 5 linearly independent eigenvectors
for A, we now know it is diagonalizable and we can build the
diagonalizing matrix.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Join[ES[1],ES[2],ES[3]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{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                        2
 
                            3 (-491 + 65 Sqrt[57])
                            ----------------------
   1                         -2589 + 343 Sqrt[57]
 
   3 (491 + 65 Sqrt[57])
   ---------------------
    2589 + 343 Sqrt[57]



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



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



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



1                        0
 


   0                        1
 


   1
:[font = input; initialization; preserveAspect]
*)
Result check:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Together[Inverse[P].A.P]
(*
:[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, 
 
    (-285 + 31*57^(1/2))/(2*57^(1/2)), 0}, 
 
   {0, 0, 0, 0, (285 + 31*57^(1/2))/(2*57^(1/2))}}]
;[o]


-2                   0
 


   0                    0
 


   0



0                    -2
 


   0                    0
 


   0



0                    0
 


   -2                   0
 


   0



0                    0
 
                        -285 + 31 Sqrt[57]
                        ------------------
   0                        2 Sqrt[57]
 


   0



0                    0
 


   0                    0
 
   285 + 31 Sqrt[57]
   -----------------
      2 Sqrt[57]
:[font = input; initialization; preserveAspect]
*)
Exercise 3.2.2
(*
:[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]
*)
lambda=x/.Solve[Det[x*IdentityMatrix[4]-A]==0,x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0}
;[o]
{0, 0, 0, 0}
:[font = input; initialization; preserveAspect; startGroup]
*)
ES[1]=NullSpace[lambda[[1]]IdentityMatrix[4]-A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 1}}]
;[o]
0   0   0   1
:[font = input; initialization; preserveAspect]
*)
There is only one eigenvector (4 needed), so this
matrix is NOT diagonalizable.
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 3.2.3
(*
:[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]
*)
lambda=x/.Solve[Det[x*IdentityMatrix[5]-A]==0,x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{1, 1, 1, 1, 6}
;[o]
{1, 1, 1, 1, 6}
:[font = input; initialization; preserveAspect; startGroup]
*)
ES[1]=NullSpace[lambda[[1]]IdentityMatrix[5]-A]
(*
:[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}}]
;[o]
-1   0    0    0    1

-1   0    0    1    0

-1   0    1    0    0

-1   1    0    0    0
:[font = input; initialization; preserveAspect; startGroup]
*)
ES[2]=NullSpace[lambda[[5]]IdentityMatrix[5]-A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 1, 1, 1, 1}}]
;[o]
1   1   1   1   1
:[font = input; initialization; preserveAspect]
*)
Diagonalizable (5 linearly independent eigenvectors).
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Join[ES[1],ES[2]]]
(*
:[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]
*)
Result check:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
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 3.2.4
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[i+j-1,{i,4},{j,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2, 3, 4}, {2, 3, 4, 5}, {3, 4, 5, 6}, 
 
   {4, 5, 6, 7}}]
;[o]
1   2   3   4

2   3   4   5

3   4   5   6

4   5   6   7
:[font = input; initialization; preserveAspect; startGroup]
*)
lambda=x/.Solve[Det[x*IdentityMatrix[4]-A]==0,x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, (16 - 4*21^(1/2))/2, (16 + 4*21^(1/2))/2}
;[o]
       16 - 4 Sqrt[21]  16 + 4 Sqrt[21]
{0, 0, ---------------, ---------------}
              2                2
:[font = input; initialization; preserveAspect; startGroup]
*)
ES[1]=NullSpace[lambda[[1]]IdentityMatrix[4]-A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, -3, 0, 1}, {1, -2, 1, 0}}]
;[o]
2    -3   0    1

1    -2   1    0
:[font = input; initialization; preserveAspect; startGroup]
*)
ES[2]=Together[NullSpace[lambda[[3]]IdentityMatrix[4]-A]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(-13 + 3*21^(1/2))/(2*(-14 + 3*21^(1/2))), 
 
    (-9 + 2*21^(1/2))/(-14 + 3*21^(1/2)), 
 
    (-23 + 5*21^(1/2))/(2*(-14 + 3*21^(1/2))), 1}}]
;[o]
  -13 + 3 Sqrt[21]     -9 + 2 Sqrt[21]
--------------------   ----------------
2 (-14 + 3 Sqrt[21])   -14 + 3 Sqrt[21]
 
     -23 + 5 Sqrt[21]
   --------------------
   2 (-14 + 3 Sqrt[21])   1
:[font = input; initialization; preserveAspect; startGroup]
*)
ES[3]=Together[NullSpace[lambda[[4]]IdentityMatrix[4]-A]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(13 + 3*21^(1/2))/(2*(14 + 3*21^(1/2))), 
 
    (9 + 2*21^(1/2))/(14 + 3*21^(1/2)), 
 
    (23 + 5*21^(1/2))/(2*(14 + 3*21^(1/2))), 1}}]
;[o]
  13 + 3 Sqrt[21]     9 + 2 Sqrt[21]
-------------------   ---------------
2 (14 + 3 Sqrt[21])   14 + 3 Sqrt[21]
 
     23 + 5 Sqrt[21]
   -------------------
   2 (14 + 3 Sqrt[21])   1
:[font = input; initialization; preserveAspect]
*)
Diagonalizable (4 linearly independent eigenvectors).
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Join[ES[1],ES[2],ES[3]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 1, (-13 + 3*21^(1/2))/
 
     (2*(-14 + 3*21^(1/2))), 
 
    (13 + 3*21^(1/2))/(2*(14 + 3*21^(1/2)))}, 
 
   {-3, -2, (-9 + 2*21^(1/2))/(-14 + 3*21^(1/2)), 
 
    (9 + 2*21^(1/2))/(14 + 3*21^(1/2))}, 
 
   {0, 1, (-23 + 5*21^(1/2))/(2*(-14 + 3*21^(1/2))), 
 
    (23 + 5*21^(1/2))/(2*(14 + 3*21^(1/2)))}, {1, 0, 1, 1}}]
;[o]


2                      1
 
     -13 + 3 Sqrt[21]       13 + 3 Sqrt[21]
   --------------------   -------------------
   2 (-14 + 3 Sqrt[21])   2 (14 + 3 Sqrt[21])



-3                     -2
 
   -9 + 2 Sqrt[21]        9 + 2 Sqrt[21]
   ----------------       ---------------
   -14 + 3 Sqrt[21]       14 + 3 Sqrt[21]



0                      1
 
     -23 + 5 Sqrt[21]       23 + 5 Sqrt[21]
   --------------------   -------------------
   2 (-14 + 3 Sqrt[21])   2 (14 + 3 Sqrt[21])



1                      0
 


   1                      1
:[font = input; initialization; preserveAspect]
*)
Result check:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Together[Inverse[P].A.P]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0}, {0, 0, 0, 0}, 
 
   {0, 0, (2*(-21 + 4*21^(1/2)))/21^(1/2), 0}, 
 
   {0, 0, 0, (2*(21 + 4*21^(1/2)))/21^(1/2)}}]
;[o]


0                      0
 


   0                      0



0                      0
 


   0                      0



0                      0
 
   2 (-21 + 4 Sqrt[21])
   --------------------
         Sqrt[21]         0



0                      0
 
                          2 (21 + 4 Sqrt[21])
                          -------------------
   0                           Sqrt[21]
:[font = input; initialization; preserveAspect]
*)
Exercise 3.2.5
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[i+j,{i,4},{j,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 3, 4, 5}, {3, 4, 5, 6}, {4, 5, 6, 7}, 
 
   {5, 6, 7, 8}}]
;[o]
2   3   4   5

3   4   5   6

4   5   6   7

5   6   7   8
:[font = input; initialization; preserveAspect]
*)
Note that Min[i,j]+Max[i,j]=i+j.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
lambda=x/.Solve[Det[x*IdentityMatrix[4]-A]==0,x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, (20 - 4*30^(1/2))/2, (20 + 4*30^(1/2))/2}
;[o]
       20 - 4 Sqrt[30]  20 + 4 Sqrt[30]
{0, 0, ---------------, ---------------}
              2                2
:[font = input; initialization; preserveAspect; startGroup]
*)
ES[1]=NullSpace[lambda[[1]]IdentityMatrix[4]-A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, -3, 0, 1}, {1, -2, 1, 0}}]
;[o]
2    -3   0    1

1    -2   1    0
:[font = input; initialization; preserveAspect; startGroup]
*)
ES[2]=Together[NullSpace[lambda[[3]]IdentityMatrix[4]-A]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(-145 + 27*30^(1/2))/(-265 + 48*30^(1/2)), 
 
    (-185 + 34*30^(1/2))/(-265 + 48*30^(1/2)), 
 
    (-225 + 41*30^(1/2))/(-265 + 48*30^(1/2)), 1}}]
;[o]
-145 + 27 Sqrt[30]   -185 + 34 Sqrt[30]
------------------   ------------------
-265 + 48 Sqrt[30]   -265 + 48 Sqrt[30]
 
   -225 + 41 Sqrt[30]
   ------------------
   -265 + 48 Sqrt[30]   1
:[font = input; initialization; preserveAspect; startGroup]
*)
ES[3]=Together[NullSpace[lambda[[4]]IdentityMatrix[4]-A]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(145 + 27*30^(1/2))/(265 + 48*30^(1/2)), 
 
    (185 + 34*30^(1/2))/(265 + 48*30^(1/2)), 
 
    (225 + 41*30^(1/2))/(265 + 48*30^(1/2)), 1}}]
;[o]
145 + 27 Sqrt[30]   185 + 34 Sqrt[30]   225 + 41 Sqrt[30]
-----------------   -----------------   -----------------
265 + 48 Sqrt[30]   265 + 48 Sqrt[30]   265 + 48 Sqrt[30]
 


      1
:[font = input; initialization; preserveAspect]
*)
Diagonalizable (4 linearly independent eigenvectors).
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Join[ES[1],ES[2],ES[3]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 1, (-145 + 27*30^(1/2))/
 
     (-265 + 48*30^(1/2)), 
 
    (145 + 27*30^(1/2))/(265 + 48*30^(1/2))}, 
 
   {-3, -2, (-185 + 34*30^(1/2))/(-265 + 48*30^(1/2)), 
 
    (185 + 34*30^(1/2))/(265 + 48*30^(1/2))}, 
 
   {0, 1, (-225 + 41*30^(1/2))/(-265 + 48*30^(1/2)), 
 
    (225 + 41*30^(1/2))/(265 + 48*30^(1/2))}, {1, 0, 1, 1}}]
;[o]


2                    1
 
   -145 + 27 Sqrt[30]   145 + 27 Sqrt[30]
   ------------------   -----------------
   -265 + 48 Sqrt[30]   265 + 48 Sqrt[30]



-3                   -2
 
   -185 + 34 Sqrt[30]   185 + 34 Sqrt[30]
   ------------------   -----------------
   -265 + 48 Sqrt[30]   265 + 48 Sqrt[30]



0                    1
 
   -225 + 41 Sqrt[30]   225 + 41 Sqrt[30]
   ------------------   -----------------
   -265 + 48 Sqrt[30]   265 + 48 Sqrt[30]



1                    0
 


   1                    1
:[font = input; initialization; preserveAspect]
*)
Result check:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Together[Inverse[P].A.P]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0}, {0, 0, 0, 0}, 
 
   {0, 0, (10/3)^(1/2)*(-6 + 30^(1/2)), 0}, 
 
   {0, 0, 0, (10/3)^(1/2)*(6 + 30^(1/2))}}]
;[o]


0                          0
 


   0                          0



0                          0
 


   0                          0



0                          0
 
        10
   Sqrt[--] (-6 + Sqrt[30])
        3                     0



0                          0
 
                                   10
                              Sqrt[--] (6 + Sqrt[30])
   0                               3
^*)