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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 203 24 "Jordan Canonical Form II"
 }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 16 "By R
ussell Blyth" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "restart: with(LinearAlgebra):" }}}{SECT 1 {PARA 
3 "" 0 "" {TEXT 205 45 "Computing the Jordan Canonical Form and Basis"
 }}{EXCHG {PARA 0 "" 0 "" {TEXT 204 23 "We tackle two examples." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "A := Matrix([[-41, -21, -11
, 126, 63, -19], [587, 118, 46, -726, -363, 292], [-60, -11, -7, 66, 3
3, -30], [35, 20, 10, -118, -60, 15], [117, -3, -6, 8, 6, 63], [94, 43
, 23, -258, -129, 44]]);\n" }{MPLTEXT 1 0 38 "factor(CharacteristicPol
ynomial(A,t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 315 "A has two dis
tinct eigenvalues. Let lambda_1 = -3 and lambda_2 = 2. We wish to find
 the dot diagram for A corresponding to each eigenvalue. Compute a bas
is for the null space of powers of (A-lamba_i*I) until the nullity mat
ches the multiplicity of the eigenvalue. For the eigenvalue -3 we are \+
looking for nullity 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "I
d := IdentityMatrix(6);\nN" }{MPLTEXT 1 0 17 "ullSpace(A+3*Id);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 204 233 "We try the next higher power of \+
A-lambda*I. We do, however, know now that since there is only one vect
or in the basis for the eigenspace, there is just one cycle to be comp
uted, which must therefore have length 2. The dot diagram is:" }}
{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 4 "\245" }}
{PARA 0 "" 0 "" {TEXT 204 4 "\245" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "betam3 := NullSpace((A+3*Id
)^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 184 "We take as our end vec
tor of the cycle the basis vector not lying in the eigenspace (if need
 be )(if TmlIv1 computes to the zero vector, change the subscript in t
he following command)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "v
1 := betam3[2];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 40 "Compute the i
nitial vector of the cycle." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "TmlIv1 := (A+3*Id).v1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 66 "We
 now have our basis beta_1 for the first generalized eigenspace:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "beta_1 := [TmlIv1, v1];" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 204 49 "We turn now to the second general
ized eigenspace." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "E2 := N
ullSpace(A-2*Id);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 104 "This tells
 us that there are two cycles, and hence two Jordan blocks, correspond
ing to the eigenvalue 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 
"NullSpace((A-2*Id)^2);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 220 "No
w we have three independent generalized eigenvectors, still not enough
 to span the generalized eigenspace corresponding to the eigenvalue 2.
 However, we are just one vector short, so we know the next step will \+
suffice." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "betap2 := NullS
pace((A-2*Id)^3);" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 206 100 "We thus \+
have found the dot diagram for the generalized eigenspace correspondin
g to the eigenvalue 2:" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 
0 "" {TEXT 204 15 "\245       \245" }}{PARA 0 "" 0 "" {TEXT 204 4 "\24
5" }}{PARA 0 "" 0 "" {TEXT 204 4 "\245" }}{PARA 0 "" 0 "" {TEXT 204 0 
"" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 200 131 
"Find a vector in this basis which will span a cycle of length three, \+
that is find one which is not in the null space of (A-2*I)^2. " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "((A-2*Id)^2).betap2[1];" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 204 91 "(If necessary, replace the index \+
1 with another index until the vector computed is nonzero)" }}{PARA 0 
"" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 26 "Compute the cyc
le vectors:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v2 := betap2
[1];\n" }{MPLTEXT 1 0 24 "TmlIv2 := (A-2*Id).v2;\n" }{MPLTEXT 1 0 27 "
TmlI2v2 := ((A-2*Id)^2).v2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 117 "
We have our first cycle of generalized eigenvectors for the generalize
d eigenspace corresponding to the eigenvalue 2." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 32 "beta_2 := [TmlI2v2, TmlIv2, v2];" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 204 353 "We still have one eigenvector to find, \+
for the remaining cycle of length 1. We should be able to find a vecto
r in the basis for the eigenspace which is not in the span of the cycl
e we already have computed. Check by finding the rank of the matrix wi
th columns the vectors from the known cycle and final column a vector \+
from the basis for the eigenspace." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "testmat := <TmlI2v2|TmlIv2|v2|E2[1]>;\n" }{MPLTEXT 1 
0 16 "Rank(testmat);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 72 "We are
 in business - we have found a vector for our cycle of length one:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "v3 := E2[1];" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 204 43 "We can now put together our basis beta f
or " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetti
ng:-mverbatim(\"6#*$)I\"RG6\"\"\"'\"\"\"\"), Typesetting:-mi(\"\"));" 
"-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'
-I*mverbatimGF$6#Q26#*$)I\"RG6\"\"\"'\"\"\"F'F+" }{TEXT 204 148 ", wit
h respect to which the limear operator L_A in is Jordan canonical form
. Our Jordan basis is the union of the three cycles we found. Try it o
ut:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Q := <TmlIv1|v1|TmlI
2v2|TmlIv2|v2|v3>;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 59 "This matri
x then transforms A to its Jordan canonical form." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 11 "Q^(-1).A.Q;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
 204 81 "For our next example, consider the vector space V which is th
e span of the set \n" }{TEXT 204 2 "\{" }{XPPEDIT 18 0 "Typesetting:-m
row(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"e\"), T
ypesetting:-mi(\"x\"), superscriptshift = \"0\"), Typesetting:-mo(\",
\", form = \"infix\", fence = \"false\", separator = \"true\", lspace \+
= \"0em\", rspace = \"verythickmathspace\", stretchy = \"false\", symm
etric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \+
\"false\", movablelimits = \"false\", accent = \"false\", font_style_n
ame = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backgro
und = \"[255,255,255]\"), Typesetting:-msup(Typesetting:-mi(\"e\"), Ty
pesetting:-mi(\"y\"), superscriptshift = \"0\"), Typesetting:-mo(\",\"
, form = \"infix\", fence = \"false\", separator = \"true\", lspace = \+
\"0em\", rspace = \"verythickmathspace\", stretchy = \"false\", symmet
ric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"
false\", movablelimits = \"false\", accent = \"false\", font_style_nam
e = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backgroun
d = \"[255,255,255]\"), Typesetting:-msup(Typesetting:-mi(\"xe\"), Typ
esetting:-mi(\"x\"), superscriptshift = \"0\"), Typesetting:-mo(\",\",
 form = \"infix\", fence = \"false\", separator = \"true\", lspace = \+
\"0em\", rspace = \"verythickmathspace\", stretchy = \"false\", symmet
ric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"
false\", movablelimits = \"false\", accent = \"false\", font_style_nam
e = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backgroun
d = \"[255,255,255]\"), Typesetting:-msup(Typesetting:-mi(\"xe\"), Typ
esetting:-mi(\"y\"), superscriptshift = \"0\"), Typesetting:-mo(\",\",
 form = \"infix\", fence = \"false\", separator = \"true\", lspace = \+
\"0em\", rspace = \"verythickmathspace\", stretchy = \"false\", symmet
ric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"
false\", movablelimits = \"false\", accent = \"false\", font_style_nam
e = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backgroun
d = \"[255,255,255]\"), Typesetting:-msup(Typesetting:-mi(\"ye\"), Typ
esetting:-mi(\"x\"), superscriptshift = \"0\"), Typesetting:-mo(\",\",
 form = \"infix\", fence = \"false\", separator = \"true\", lspace = \+
\"0em\", rspace = \"verythickmathspace\", stretchy = \"false\", symmet
ric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"
false\", movablelimits = \"false\", accent = \"false\", font_style_nam
e = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backgroun
d = \"[255,255,255]\"), Typesetting:-msup(Typesetting:-mi(\"ye\"), Typ
esetting:-mi(\"y\"), superscriptshift = \"0\"), Typesetting:-mi(\"\"))
;" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6/-I#miGF$6#Q
!F'-I%msupGF$6%-F,6#Q\"eF'-F,6#Q\"xF'/%1superscriptshiftGQ\"0F'-I#moGF
$63Q\",F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%*separatorGQ%trueF'/%'ls
paceGQ$0emF'/%'rspaceGQ3verythickmathspaceF'/%)stretchyGFD/%*symmetric
GFD/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopGFD/%.movablelimi
tsGFD/%'accentGFD/%0font_style_nameGQ+2D~CommentF'/%%sizeGQ#12F'/%+for
egroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F'-F06%F2-F,6#Q\"yF'F8
F;-F06%-F,6#Q#xeF'F5F8F;-F06%F[pFfoF8F;-F06%-F,6#Q#yeF'F5F8F;-F06%FbpF
foF8F+" }{TEXT 204 106 "\} of functions in the two variables x and y, \+
and suppose T is the linear operator defined by T(f(x,y)) = " }
{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m
row(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mrow(Typeset
ting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"&PartialD;\")), Ty
pesetting:-mi(\"\")), Typesetting:-mn(\"2\"), superscriptshift = \"0\"
), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modu
lenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%F+-I%msupGF$6
%-F#6%F+-F#6#-F,6#Q+&PartialD;F'F+-I#mnGF$6#Q\"2F'/%1superscriptshiftG
Q\"0F'F+F+" }{TEXT 204 87 "f/\266x\266y. If possible, let's find the J
ordan form of T, and a Jordan basis for T.\n" }}{PARA 0 "" 0 "" {TEXT 
204 66 "First, let's find the matrix for T relative to the basis provi
ded." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "diff(e^x,x,y);\n" }
{MPLTEXT 1 0 16 "diff(e^y,x,y);\n" }{MPLTEXT 1 0 18 "diff(x*e^x,x,y);
\n" }{MPLTEXT 1 0 18 "diff(x*e^y,x,y);\n" }{MPLTEXT 1 0 18 "diff(y*e^x
,x,y);\n" }{MPLTEXT 1 0 16 "diff(y*e^y,x,y);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 204 91 "Since ln(e) = 1  we get the matrix for T relative to t
he basis provided to be the matrix B:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "B := Matrix([[0,0,0,0,1,0],\n" }{MPLTEXT 1 0 32 " [0,
0,0,1,0,0], [0,0,0,0,0,0],\n" }{MPLTEXT 1 0 32 " [0,0,0,0,0,0], [0,0,0
,0,0,0],\n" }{MPLTEXT 1 0 19 " [0,0,0,0,0,0]]);\n" }{MPLTEXT 1 0 38 "f
actor(CharacteristicPolynomial(B,t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
 204 127 "Hence there is precisely one eigenvalue, namely 0, for T. Le
t's find the first power of T ( = (T-0I) ) that annihilates all of " }
{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m
row(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"R\"), T
ypesetting:-mn(\"6\"), superscriptshift = \"0\"), Typesetting:-mi(\"\"
)), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesetting
GI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%F+-I%msupGF$6%-F,6#Q\"RF'-I#mnGF$6#Q
\"6F'/%1superscriptshiftGQ\"0F'F+F+" }{TEXT 204 265 ". Certainly T its
elf does not; note also that T has rank 2, and hence nullity 4, so tha
t the eigenspace of T corresponding to the eigenvalue 0 has dimension \+
4 - hence there are 4 Jordan blocks to be determined. We continue with
 the computation of the null space of " }{XPPEDIT 18 0 "Typesetting:-m
row(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Ty
pesetting:-msup(Typesetting:-mi(\"T\"), Typesetting:-mn(\"2\"), supers
criptshift = \"0\"), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" 
"-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'
-F#6%F+-I%msupGF$6%-F,6#Q\"TF'-I#mnGF$6#Q\"2F'/%1superscriptshiftGQ\"0
F'F+F+" }{TEXT 204 5 " ( = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typeset
ting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-
msup(Typesetting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence
 = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace
 = \"thinmathspace\", stretchy = \"true\", symmetric = \"false\", maxs
ize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimit
s = \"false\", accent = \"false\", font_style_name = \"2D Comment\", s
ize = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"
), Typesetting:-mrow(Typesetting:-mi(\"t\"), Typesetting:-mo(\"&minus;
\", form = \"infix\", fence = \"false\", separator = \"false\", lspace
 = \"mediummathspace\", rspace = \"mediummathspace\", stretchy = \"fal
se\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", \+
largeop = \"false\", movablelimits = \"false\", accent = \"false\", fo
nt_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]
\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mn
(\"0\"), Typesetting:-mo(\"&InvisibleTimes;\", form = \"infix\", fence
 = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em
\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity
\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a
ccent = \"false\", font_style_name = \"2D Comment\", size = \"12\", fo
reground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-
mi(\"I\")), Typesetting:-mi(\"\")), Typesetting:-mo(\")\", form = \"po
stfix\", fence = \"true\", separator = \"false\", lspace = \"thinmaths
pace\", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric
 = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fal
se\", movablelimits = \"false\", accent = \"false\", font_style_name =
 \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background =
 \"[255,255,255]\")), Typesetting:-mn(\"2\"), superscriptshift = \"0\"
), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modu
lenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%F+-I%msupGF$6
%-F#6%-I#moGF$63Q\"(F'/%%formGQ'prefixF'/%&fenceGQ%trueF'/%*separatorG
Q&falseF'/%'lspaceGQ.thinmathspaceF'/%'rspaceGFE/%)stretchyGF?/%*symme
tricGFB/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopGFB/%.movable
limitsGFB/%'accentGFB/%0font_style_nameGQ+2D~CommentF'/%%sizeGQ#12F'/%
+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F'-F#6&-F,6#Q\"tF'
-F763Q(&minus;F'/F;Q&infixF'/F>FBF@/FDQ0mediummathspaceF'/FGFjo/FIFBFJ
FLFOFRFTFVFXFenFhnF[o-F#6%-I#mnGF$6#Q\"0F'-F763Q1&InvisibleTimes;F'Ffo
FhoF@/FDQ$0emF'/FGFgpF\\pFJFLFOFRFTFVFXFenFhnF[o-F,6#Q\"IF'F+-F763Q\")
F'/F;Q(postfixF'F=F@FC/FGQ2verythinmathspaceF'FHFJFLFOFRFTFVFXFenFhnF[
o-F`p6#Q\"2F'/%1superscriptshiftGQ\"0F'F+F+" }{TEXT 204 2 ")." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "NullSpace(B^2);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 204 65 "Note that this null space has dimension \+
6, and hence annihilates " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetti
ng:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-ms
up(Typesetting:-mi(\"R\"), Typesetting:-mn(\"6\"), superscriptshift = \+
\"0\"), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I
+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%F+-I%msu
pGF$6%-F,6#Q\"RF'-I#mnGF$6#Q\"6F'/%1superscriptshiftGQ\"0F'F+F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 204 30 "Thus the dot diagram for T is:" }
}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 37 "\245  \+
     \245       \245       \245" }}{PARA 0 "" 0 "" {TEXT 204 15 "\245 \+
      \245" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 
204 143 "We continue by looking for vectors which are not annihilated \+
by T - actually, the matrix B itself gives a direct hint - two such ve
ctors are  x" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"),
 Typesetting:-msup(Typesetting:-mi(\"e\"), Typesetting:-mi(\"y\"), sup
erscriptshift = \"0\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulen
ameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msupGF$6%-F,6#Q\"e
F'-F,6#Q\"yF'/%1superscriptshiftGQ\"0F'F+" }{TEXT 204 8 "  and  y" }
{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m
sup(Typesetting:-mi(\"e\"), Typesetting:-mi(\"x\"), superscriptshift =
 \"0\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese
ttingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msupGF$6%-F,6#Q\"eF'-F,6#Q\"xF'/%
1superscriptshiftGQ\"0F'F+" }{TEXT 204 6 ".  T(x" }{XPPEDIT 18 0 "Type
setting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi
(\"e\"), Typesetting:-mi(\"y\"), superscriptshift = \"0\"), Typesettin
g:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6
%-I#miGF$6#Q!F'-I%msupGF$6%-F,6#Q\"eF'-F,6#Q\"yF'/%1superscriptshiftGQ
\"0F'F+" }{TEXT 204 4 ") = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typeset
ting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"e\"), Typesetting:
-mi(\"y\"), superscriptshift = \"0\"), Typesetting:-mi(\"\"));" "-I%mr
owG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msu
pGF$6%-F,6#Q\"eF'-F,6#Q\"yF'/%1superscriptshiftGQ\"0F'F+" }{TEXT 204 
21 " (giving the cycle \{" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetti
ng:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"e\"), Typesetting:-m
i(\"y\"), superscriptshift = \"0\"), Typesetting:-mi(\"\"));" "-I%mrow
G6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msupG
F$6%-F,6#Q\"eF'-F,6#Q\"yF'/%1superscriptshiftGQ\"0F'F+" }{TEXT 204 3 "
, x" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesett
ing:-msup(Typesetting:-mi(\"e\"), Typesetting:-mi(\"y\"), superscripts
hift = \"0\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,
TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msupGF$6%-F,6#Q\"eF'-F,6#Q
\"yF'/%1superscriptshiftGQ\"0F'F+" }{TEXT 204 15 "\}), while  T(y" }
{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m
sup(Typesetting:-mi(\"e\"), Typesetting:-mi(\"x\"), superscriptshift =
 \"0\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese
ttingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msupGF$6%-F,6#Q\"eF'-F,6#Q\"xF'/%
1superscriptshiftGQ\"0F'F+" }{TEXT 204 4 ") = " }{XPPEDIT 18 0 "Typese
tting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(
\"e\"), Typesetting:-mi(\"x\"), superscriptshift = \"0\"), Typesetting
:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%
-I#miGF$6#Q!F'-I%msupGF$6%-F,6#Q\"eF'-F,6#Q\"xF'/%1superscriptshiftGQ
\"0F'F+" }{TEXT 204 21 " (giving the cycle \{" }{XPPEDIT 18 0 "Typeset
ting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"
e\"), Typesetting:-mi(\"x\"), superscriptshift = \"0\"), Typesetting:-
mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I
#miGF$6#Q!F'-I%msupGF$6%-F,6#Q\"eF'-F,6#Q\"xF'/%1superscriptshiftGQ\"0
F'F+" }{TEXT 204 3 ", y" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting
:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"e\"), Typesetting:-mi(
\"x\"), superscriptshift = \"0\"), Typesetting:-mi(\"\"));" "-I%mrowG6
#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msupGF$
6%-F,6#Q\"eF'-F,6#Q\"xF'/%1superscriptshiftGQ\"0F'F+" }{TEXT 204 180 "
\}).  The other members of the original basis are all eigenvectors cor
responding to the eigenvalue 0, and hence if we simply re-order the or
iginal spanning basis as follows:    \{ " }{XPPEDIT 18 0 "Typesetting:
-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"e\"),
 Typesetting:-mi(\"y\"), superscriptshift = \"0\"), Typesetting:-mi(\"
\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF
$6#Q!F'-I%msupGF$6%-F,6#Q\"eF'-F,6#Q\"yF'/%1superscriptshiftGQ\"0F'F+"
 }{TEXT 204 3 ", x" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(
\"\"), Typesetting:-msup(Typesetting:-mi(\"e\"), Typesetting:-mi(\"y\"
), superscriptshift = \"0\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+m
odulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msupGF$6%-F,
6#Q\"eF'-F,6#Q\"yF'/%1superscriptshiftGQ\"0F'F+" }{TEXT 204 2 ", " }
{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m
sup(Typesetting:-mi(\"e\"), Typesetting:-mi(\"x\"), superscriptshift =
 \"0\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese
ttingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msupGF$6%-F,6#Q\"eF'-F,6#Q\"xF'/%
1superscriptshiftGQ\"0F'F+" }{TEXT 204 3 ", y" }{XPPEDIT 18 0 "Typeset
ting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"
e\"), Typesetting:-mi(\"x\"), superscriptshift = \"0\"), Typesetting:-
mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I
#miGF$6#Q!F'-I%msupGF$6%-F,6#Q\"eF'-F,6#Q\"xF'/%1superscriptshiftGQ\"0
F'F+" }{TEXT 204 3 ", x" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting
:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"e\"), Typesetting:-mi(
\"x\"), superscriptshift = \"0\"), Typesetting:-mi(\"\"));" "-I%mrowG6
#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msupGF$
6%-F,6#Q\"eF'-F,6#Q\"xF'/%1superscriptshiftGQ\"0F'F+" }{TEXT 204 3 ", \+
y" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesettin
g:-msup(Typesetting:-mi(\"e\"), Typesetting:-mi(\"y\"), superscriptshi
ft = \"0\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Ty
pesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msupGF$6%-F,6#Q\"eF'-F,6#Q\"y
F'/%1superscriptshiftGQ\"0F'F+" }{TEXT 204 153 " \}, then the matrix o
f  T  relative to this new basis is in the following Jordan form.  Not
e there are two 2x2 Jordan blocks, and two 1x1 Jordan blocks." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "Matrix([[0, 1, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0,
 0, 0, 0], [0, 0, 0, 0, 0, 0]]);" }}}{SECT 1 {PARA 211 "" 0 "" {TEXT 
203 8 "Exercise" }}{PARA 0 "" 0 "" {TEXT 200 102 "Find a Jordan basis \+
for the matrix C, using an analysis similar to the one performed for t
he matrix A." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "C := Matrix
([[1, 1, -25, -1, 0, 0, 0, 14], [3, -2, 28, -3, 1, 1, 3, -18], [0, 0, \+
1, 1, 0, 0, -1, 0], [0, -3, 27, 1, 1, 0, 0, -18], [9, -9, 84, 0, 4, 3,
 0, -54], [0, -3, 75, 3, 0, 1, 1, -42], [0, 0, -2, 0, 0, 0, 1, 1], [0,
 0, 0, 2, 0, 0, -2, 1]]);" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "
" "%#%?G" }}}}}}
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