diff --git a/notes/jordan-vectors.lyx b/notes/jordan-vectors.lyx
index 8965f0f..50858a1 100644
--- a/notes/jordan-vectors.lyx
+++ b/notes/jordan-vectors.lyx
@@ -1,5 +1,5 @@
-#LyX 2.2 created this file. For more info see http://www.lyx.org/
-\lyxformat 508
+#LyX 2.4 created this file. For more info see https://www.lyx.org/
+\lyxformat 620
 \begin_document
 \begin_header
 \save_transient_properties true
@@ -11,11 +11,11 @@
 \date{Created Spring 2009; updated \today}
 \end_preamble
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+\maintain_unincluded_children no
 \language english
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 \font_typewriter "default" "default"
@@ -23,9 +23,13 @@
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 \font_sf_scale 100 100
 \font_tt_scale 100 100
+\use_microtype false
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 \graphics default
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 \output_sync 0
@@ -55,6 +59,9 @@
 \suppress_date false
 \justification true
 \use_refstyle 0
+\use_formatted_ref 0
+\use_minted 0
+\use_lineno 0
 \index Index
 \shortcut idx
 \color #008000
@@ -63,21 +70,30 @@
 \tocdepth 3
 \paragraph_separation indent
 \paragraph_indentation default
-\quotes_language english
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style english
+\dynamic_quotes 0
 \papercolumns 2
 \papersides 2
 \paperpagestyle default
+\tablestyle default
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 \html_math_output 0
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+\docbook_mathml_prefix 1
 \end_header
 
 \begin_body
 
 \begin_layout Title
-A useful basis for defective matrices: 
+A useful basis for defective matrices:
+ 
 \begin_inset Newline newline
 \end_inset
 
@@ -87,7 +103,8 @@ Jordan vectors and the Jordan form
 \begin_layout Author
 S.
  G.
- Johnson, MIT 18.06
+ Johnson,
+ MIT 18.06
 \end_layout
 
 \begin_layout Abstract
@@ -124,14 +141,14 @@ Jordan chains
 \end_inset
 
 ).
- In these notes, instead, I omit most of the formal derivations and instead
- focus on the 
+ In these notes,
+ instead,
+ I omit most of the formal derivations and instead focus on the 
 \emph on
 consequences
 \emph default
  of the Jordan vectors for how we understand matrices.
- What happens to our traditional eigenvector-based pictures of things like
- 
+ What happens to our traditional eigenvector-based pictures of things like 
 \begin_inset Formula $A^{n}$
 \end_inset
 
@@ -143,11 +160,14 @@ consequences
 \begin_inset Formula $A$
 \end_inset
 
- fails? The answer, for any matrix function 
+ fails?
+ The answer,
+ for any matrix function 
 \begin_inset Formula $f(A)$
 \end_inset
 
-, turns out to involve the 
+,
+ turns out to involve the 
 \emph on
 derivative
 \emph default
@@ -163,7 +183,8 @@ Introduction
 \end_layout
 
 \begin_layout Standard
-So far in the eigenproblem portion of 18.06, our strategy has been simple:
+So far in the eigenproblem portion of 18.06,
+ our strategy has been simple:
  find the eigenvalues 
 \begin_inset Formula $\lambda_{i}$
 \end_inset
@@ -176,7 +197,8 @@ So far in the eigenproblem portion of 18.06, our strategy has been simple:
 \begin_inset Formula $A$
 \end_inset
 
-, expand any vector of interest 
+,
+ expand any vector of interest 
 \begin_inset Formula $\vec{u}$
 \end_inset
 
@@ -184,7 +206,8 @@ So far in the eigenproblem portion of 18.06, our strategy has been simple:
 \begin_inset Formula $\vec{u}=c_{1}\vec{x}_{1}+\cdots+c_{n}\vec{x}_{n})$
 \end_inset
 
-, and then any operation with 
+,
+ and then any operation with 
 \begin_inset Formula $A$
 \end_inset
 
@@ -193,7 +216,8 @@ So far in the eigenproblem portion of 18.06, our strategy has been simple:
 \end_inset
 
  acting on each eigenvector.
- So, 
+ So,
+ 
 \begin_inset Formula $A^{k}$
 \end_inset
 
@@ -201,7 +225,8 @@ So far in the eigenproblem portion of 18.06, our strategy has been simple:
 \begin_inset Formula $\lambda_{i}^{k}$
 \end_inset
 
-, 
+,
+ 
 \begin_inset Formula $e^{At}$
 \end_inset
 
@@ -209,8 +234,10 @@ So far in the eigenproblem portion of 18.06, our strategy has been simple:
 \begin_inset Formula $e^{\lambda_{i}t}$
 \end_inset
 
-, and so on.
- But this relied on one key assumption: we require the 
+,
+ and so on.
+ But this relied on one key assumption:
+ we require the 
 \begin_inset Formula $n\times n$
 \end_inset
 
@@ -239,7 +266,8 @@ diagonalizable
 \end_layout
 
 \begin_layout Standard
-Many important cases are always diagonalizable: matrices with 
+Many important cases are always diagonalizable:
+ matrices with 
 \begin_inset Formula $n$
 \end_inset
 
@@ -247,8 +275,9 @@ Many important cases are always diagonalizable: matrices with
 \begin_inset Formula $\lambda_{i}$
 \end_inset
 
-, real symmetric or orthogonal matrices, and complex Hermitian or unitary
- matrices.
+,
+ real symmetric or orthogonal matrices,
+ and complex Hermitian or unitary matrices.
  But there are rare cases where 
 \begin_inset Formula $A$
 \end_inset
@@ -261,12 +290,14 @@ not
 \begin_inset Formula $n$
 \end_inset
 
- eigenvectors: such matrices are called 
+ eigenvectors:
+ such matrices are called 
 \series bold
 defective
 \series default
 .
- For example, consider the matrix 
+ For example,
+ consider the matrix 
 \begin_inset Formula 
 \[
 A=\left(\begin{array}{cc}
@@ -281,24 +312,31 @@ This matrix has a characteristic polynomial
 \begin_inset Formula $\lambda^{2}-2\lambda+1$
 \end_inset
 
-, with a repeated root (a single eigenvalue) 
+,
+ with a repeated root (a single eigenvalue) 
 \begin_inset Formula $\lambda_{1}=1$
 \end_inset
 
 .
- (Equivalently, since 
+ (Equivalently,
+ since 
 \begin_inset Formula $A$
 \end_inset
 
- is upper triangular, we can read the determinant of 
+ is upper triangular,
+ we can read the determinant of 
 \begin_inset Formula $A-\lambda I$
 \end_inset
 
-, and hence the eigenvalues, off the diagonal.) However, it only has a 
+,
+ and hence the eigenvalues,
+ off the diagonal.) However,
+ it only has a 
 \emph on
 single
 \emph default
- indepenent eigenvector, because 
+ indepenent eigenvector,
+ because 
 \begin_inset Formula 
 \[
 A-I=\left(\begin{array}{cc}
@@ -309,7 +347,8 @@ A-I=\left(\begin{array}{cc}
 
 \end_inset
 
-is obviously rank 1, and has a one-dimensional nullspace spanned by 
+is obviously rank 1,
+ and has a one-dimensional nullspace spanned by 
 \begin_inset Formula $\vec{x}_{1}=(1,0)$
 \end_inset
 
@@ -317,13 +356,14 @@ is obviously rank 1, and has a one-dimensional nullspace spanned by
 \end_layout
 
 \begin_layout Standard
-Defective matrices arise rarely in practice, and usually only when one arranges
- for them intentionally, so we have not worried about them up to now.
- But it is important to have some idea of what happens when you have a defective
- matrix.
- Partially for computational purposes, but also to understand conceptually
- what is possible.
- For example, what will be the result of 
+Defective matrices arise rarely in practice,
+ and usually only when one arranges for them intentionally,
+ so we have not worried about them up to now.
+ But it is important to have some idea of what happens when you have a defective matrix.
+ Partially for computational purposes,
+ but also to understand conceptually what is possible.
+ For example,
+ what will be the result of 
 \begin_inset Formula 
 \[
 A^{k}\left(\begin{array}{c}
@@ -341,7 +381,8 @@ for the defective matrix
 \begin_inset Formula $A$
 \end_inset
 
- above, since 
+ above,
+ since 
 \begin_inset Formula $(1,2)$
 \end_inset
 
@@ -349,7 +390,9 @@ for the defective matrix
 \begin_inset Formula $A$
 \end_inset
 
-? For diagonalizable matrices, this would grow as 
+?
+ For diagonalizable matrices,
+ this would grow as 
 \begin_inset Formula $\lambda^{k}$
 \end_inset
 
@@ -357,39 +400,44 @@ for the defective matrix
 \begin_inset Formula $e^{\lambda t}$
 \end_inset
 
-, respectively, but what about defective matrices? Although matrices in
- real applications are rarely 
+,
+ respectively,
+ but what about defective matrices?
+ Although matrices in real applications are rarely 
 \emph on
 exactly
 \emph default
- defective, it sometimes happens (often by design!) that they are 
+ defective,
+ it sometimes happens (often by design!) that they are 
 \emph on
 nearly 
 \emph default
-defective, and we can think of exactly defective matrices as a limiting
- case.
+defective,
+ and we can think of exactly defective matrices as a limiting case.
  (The book 
 \emph on
 Spectra and Pseudospectra
 \emph default
- by Trefethen & Embree is a much more detailed dive into the fascinating
- world of nearly defective matrices.)
+ by Trefethen & Embree is a much more detailed dive into the fascinating world of nearly defective matrices.)
 \end_layout
 
 \begin_layout Standard
 The textbook (
 \emph on
 Intro.
- to Linear Algebra, 5th ed.
+ to Linear Algebra,
+ 5th ed.
 
 \emph default
- by Strang) covers the defective case only briefly, in section 8.3, with
- something called the 
+ by Strang) covers the defective case only briefly,
+ in section 8.3,
+ with something called the 
 \series bold
 Jordan form
 \series default
- of the matrix, a generalization of diagonalization, but in this section
- we will focus more on the 
+ of the matrix,
+ a generalization of diagonalization,
+ but in this section we will focus more on the 
 \begin_inset Quotes eld
 \end_inset
 
@@ -398,11 +446,11 @@ Jordan vectors
 \end_inset
 
  than on the Jordan factorization.
- For a diagonalizable matrix, the fundamental vectors are the eigenvectors,
- which are useful in their own right and give the diagonalization of the
- matrix as a side-effect.
- For a defective matrix, to get a complete basis we need to supplement the
- eigenvectors with something called 
+ For a diagonalizable matrix,
+ the fundamental vectors are the eigenvectors,
+ which are useful in their own right and give the diagonalization of the matrix as a side-effect.
+ For a defective matrix,
+ to get a complete basis we need to supplement the eigenvectors with something called 
 \series bold
 Jordan vectors
 \series default
@@ -411,9 +459,11 @@ Jordan vectors
 generalized eigenvectors
 \series default
 .
- Jordan vectors are useful in their own right, just like eigenvectors, and
- also give the Jordan form.
- Here, we'll focus mainly on the 
+ Jordan vectors are useful in their own right,
+ just like eigenvectors,
+ and also give the Jordan form.
+ Here,
+ we'll focus mainly on the 
 \emph on
 consequences
 \emph default
@@ -431,7 +481,8 @@ Defining
 \end_layout
 
 \begin_layout Standard
-In the example above, we had a 
+In the example above,
+ we had a 
 \begin_inset Formula $2\times2$
 \end_inset
 
@@ -449,21 +500,25 @@ In the example above, we had a
 \end_inset
 
 .
- Of course, we could pick another vector at random, as long as it is independent
- of 
+ Of course,
+ we could pick another vector at random,
+ as long as it is independent of 
 \begin_inset Formula $\vec{x}_{1}$
 \end_inset
 
-, but we'd like it to have something to do with 
+,
+ but we'd like it to have something to do with 
 \begin_inset Formula $A$
 \end_inset
 
-, in order to help us with computations just like eigenvectors.
+,
+ in order to help us with computations just like eigenvectors.
  The key thing is to look at 
 \begin_inset Formula $A-I$
 \end_inset
 
- above, and to notice that 
+ above,
+ and to notice that 
 \begin_inset Formula $(A-I)^{2}=0$
 \end_inset
 
@@ -472,7 +527,8 @@ In the example above, we had a
 \series bold
 nilpotent
 \series default
- if some power is the zero matrix.) So, the nullspace of 
+ if some power is the zero matrix.) So,
+ the nullspace of 
 \begin_inset Formula $(A-I)^{2}$
 \end_inset
 
@@ -485,11 +541,13 @@ extra
 \end_inset
 
  basis vector beyond the eigenvector.
- But this extra vector must still be related to the eigenvector! If 
+ But this extra vector must still be related to the eigenvector!
+ If 
 \begin_inset Formula $\vec{y}\in N[(A-I)^{2}]$
 \end_inset
 
-, then 
+,
+ then 
 \begin_inset Formula $(A-I)\vec{y}$
 \end_inset
 
@@ -497,7 +555,8 @@ extra
 \begin_inset Formula $N(A-I)$
 \end_inset
 
-, which means that 
+,
+ which means that 
 \begin_inset Formula $(A-I)\vec{y}$
 \end_inset
 
@@ -505,7 +564,8 @@ extra
 \begin_inset Formula $\vec{x}_{1}$
 \end_inset
 
-! We just need to find a new 
+!
+ We just need to find a new 
 \series bold
 
 \begin_inset Quotes eld
@@ -541,11 +601,13 @@ generalized eigenvector
 
 \end_inset
 
-Notice that, since 
+Notice that,
+ since 
 \begin_inset Formula $\vec{x}_{1}\in N(A-I)$
 \end_inset
 
-, we can add any multiple of 
+,
+ we can add any multiple of 
 \begin_inset Formula $\vec{x}_{1}$
 \end_inset
 
@@ -553,7 +615,8 @@ Notice that, since
 \begin_inset Formula $\vec{j}_{1}$
 \end_inset
 
- and still have a solution, so we can use Gram-Schmidt to get a 
+ and still have a solution,
+ so we can use Gram-Schmidt to get a 
 \emph on
 unique
 \emph default
@@ -570,7 +633,8 @@ unique
 \begin_inset Formula $2\times2$
 \end_inset
 
- equation is easy enough for us to solve by inspection, obtaining 
+ equation is easy enough for us to solve by inspection,
+ obtaining 
 \begin_inset Formula $\vec{j}_{1}=(0,1)$
 \end_inset
 
@@ -583,11 +647,13 @@ orthonormal
 \begin_inset Formula $\mathbb{R}^{2}$
 \end_inset
 
-, and our basis has some simple relationship to 
+,
+ and our basis has some simple relationship to 
 \begin_inset Formula $A$
 \end_inset
 
-! 
+!
+ 
 \end_layout
 
 \begin_layout Standard
@@ -595,7 +661,8 @@ Before we talk about how to
 \emph on
 use
 \emph default
- these Jordan vectors, let's give a more general definition.
+ these Jordan vectors,
+ let's give a more general definition.
  Suppose that 
 \begin_inset Formula $\lambda_{i}$
 \end_inset
@@ -608,11 +675,15 @@ use
 \begin_inset Formula $\det(A-\lambda_{i}I)$
 \end_inset
 
-, but with only a single (ordinary) eigenvector 
+,
+ but with only a single (ordinary) eigenvector 
 \begin_inset Formula $\vec{x}_{i}$
 \end_inset
 
-, satisfying, as usual: 
+,
+ satisfying,
+ as usual:
+ 
 \begin_inset Formula 
 \[
 (A-\lambda_{i}I)\vec{x}_{i}=0.
@@ -624,7 +695,8 @@ If
 \begin_inset Formula $\lambda_{i}$
 \end_inset
 
- is a double root, we will need a second vector to complete our basis.
+ is a double root,
+ we will need a second vector to complete our basis.
  Remarkably,
 \begin_inset Foot
 status collapsed
@@ -654,12 +726,14 @@ always
 \begin_inset Formula $N([A-\lambda_{i}I]^{2})$
 \end_inset
 
- is two-dimensional, just as for the 
+ is two-dimensional,
+ just as for the 
 \begin_inset Formula $2\times2$
 \end_inset
 
  example above.
- Hence, we can 
+ Hence,
+ we can 
 \emph on
 always
 \emph default
@@ -667,7 +741,8 @@ always
 \begin_inset Formula $\vec{j}_{i}$
 \end_inset
 
- satisfying: 
+ satisfying:
+ 
 \begin_inset Formula 
 \[
 \boxed{(A-\lambda_{i}I)\vec{j}_{i}=\vec{x}_{i},\qquad\vec{j}_{i}\perp\vec{x}_{i}}.
@@ -675,7 +750,8 @@ always
 
 \end_inset
 
-Again, we can choose 
+Again,
+ we can choose 
 \begin_inset Formula $\vec{j}_{i}$
 \end_inset
 
@@ -683,13 +759,16 @@ Again, we can choose
 \begin_inset Formula $\vec{x}_{i}$
 \end_inset
 
- via Gram-Schmidt—this is not strictly necessary, but gives a convenient
- orthogonal basis.
- (That is, the complete solution is always of the form 
+ via Gram-Schmidt—
+this is not strictly necessary,
+ but gives a convenient orthogonal basis.
+ (That is,
+ the complete solution is always of the form 
 \begin_inset Formula $\vec{x}_{p}+c\vec{x}_{i}$
 \end_inset
 
-, a particular solution 
+,
+ a particular solution 
 \begin_inset Formula $\vec{x}_{p}$
 \end_inset
 
@@ -764,7 +843,8 @@ A more general notation is to use
 \begin_inset Formula $\lambda_{i}$
 \end_inset
 
- is a triple root, we would find a third vector 
+ is a triple root,
+ we would find a third vector 
 \begin_inset Formula $\vec{x}_{i}^{(3)}$
 \end_inset
 
@@ -776,8 +856,10 @@ A more general notation is to use
 \begin_inset Formula $(A-\lambda_{i}I)\vec{x}_{i}^{(3)}=\vec{x}_{i}^{(2)}$
 \end_inset
 
-, and so on.
- In general, if 
+,
+ and so on.
+ In general,
+ if 
 \begin_inset Formula $\lambda_{i}$
 \end_inset
 
@@ -785,7 +867,8 @@ A more general notation is to use
 \begin_inset Formula $m$
 \end_inset
 
--times repeated root, then 
+-times repeated root,
+ then 
 \begin_inset Formula $N([A-\lambda_{i}]^{m})$
 \end_inset
 
@@ -793,8 +876,7 @@ A more general notation is to use
 \begin_inset Formula $m$
 \end_inset
 
--dimensiohnal we will always be able to find an orthogonal sequence (a Jordan
- chain) of Jordan vectors 
+-dimensiohnal we will always be able to find an orthogonal sequence (a Jordan chain) of Jordan vectors 
 \begin_inset Formula $\vec{x}_{i}^{(j)}$
 \end_inset
 
@@ -811,9 +893,11 @@ A more general notation is to use
 \end_inset
 
 .
- Even more generally, you might have cases with e.g.
- a triple root and two ordinary eigenvectors, where you need only one generalize
-d eigenvector, or an 
+ Even more generally,
+ you might have cases with e.g.
+ a triple root and two ordinary eigenvectors,
+ where you need only one generalized eigenvector,
+ or an 
 \begin_inset Formula $m$
 \end_inset
 
@@ -826,13 +910,17 @@ d eigenvector, or an
 \end_inset
 
  Jordan vectors.
- However, cases with more than a double root are extremely rare in practice.
- Defective matrices are rare enough to begin with, so here we'll stick with
- the most common defective matrix, one with a double root 
+ However,
+ cases with more than a double root are extremely rare in practice.
+ Defective matrices are rare enough to begin with,
+ so here we'll stick with the most common defective matrix,
+ one with a double root 
 \begin_inset Formula $\lambda_{i}$
 \end_inset
 
-: hence, one ordinary eigenvector 
+:
+ hence,
+ one ordinary eigenvector 
 \begin_inset Formula $\vec{x}_{i}$
 \end_inset
 
@@ -856,7 +944,8 @@ Using an eigenvector
 \begin_inset Formula $\vec{x}_{i}$
 \end_inset
 
- is easy: multiplying by 
+ is easy:
+ multiplying by 
 \begin_inset Formula $A$
 \end_inset
 
@@ -869,7 +958,8 @@ Using an eigenvector
 \begin_inset Formula $\vec{j}_{i}$
 \end_inset
 
-? The key is in the definition
+?
+ The key is in the definition
 \family roman
 \series medium
 \shape up
@@ -887,8 +977,7 @@ A\vec{j}_{i}=\lambda_{i}\vec{j}_{i}+\vec{x}_{i}.
 
 \end_inset
 
-It will turn out that this has a simple consequence for more complicated
- expressions like 
+It will turn out that this has a simple consequence for more complicated expressions like 
 \begin_inset Formula $A^{k}$
 \end_inset
 
@@ -896,7 +985,8 @@ It will turn out that this has a simple consequence for more complicated
 \begin_inset Formula $e^{At}$
 \end_inset
 
-, but that's probably not obvious yet.
+,
+ but that's probably not obvious yet.
  Let's try multiplying by 
 \begin_inset Formula $A^{2}$
 \end_inset
@@ -925,8 +1015,9 @@ It will turn out that this has a simple consequence for more complicated
 
 \end_inset
 
- From this, it's not hard to see the general pattern (which can be formally
- proved by induction): 
+ From this,
+ it's not hard to see the general pattern (which can be formally proved by induction):
+ 
 \begin_inset Formula 
 \[
 \boxed{A^{k}\vec{j}_{i}=\lambda_{i}^{k}\vec{j}_{i}+k\lambda_{i}^{k-1}\vec{x}_{i}}.
@@ -939,8 +1030,7 @@ Notice that the coefficient in the second term is exactly
 \end_inset
 
 .
- This is the clue we need to get the general formula to apply any function
- 
+ This is the clue we need to get the general formula to apply any function 
 \begin_inset Formula $f(A)$
 \end_inset
 
@@ -948,7 +1038,8 @@ Notice that the coefficient in the second term is exactly
 \begin_inset Formula $A$
 \end_inset
 
- to the eigenvector and the Jordan vector: 
+ to the eigenvector and the Jordan vector:
+ 
 \begin_inset Formula 
 \[
 f(A)\vec{x}_{i}=f(\lambda_{i})\vec{x}_{i},
@@ -968,8 +1059,9 @@ Multiplying by a function of the matrix multiplies
 \begin_inset Formula $\vec{j}_{i}$
 \end_inset
 
- by the same function of the eigenvalue, just as for an eigenvector, but
- 
+ by the same function of the eigenvalue,
+ just as for an eigenvector,
+ but 
 \family default
 \series default
 \shape default
@@ -993,7 +1085,9 @@ derivative
 \end_inset
 
 .
- So, for example: 
+ So,
+ for example:
+ 
 \begin_inset Formula 
 \[
 \boxed{e^{At}\vec{j}_{i}=e^{\lambda_{i}t}\vec{j}_{i}+te^{\lambda_{i}t}\vec{x}_{i}}.
@@ -1001,8 +1095,7 @@ derivative
 
 \end_inset
 
- We can show this explicitly by considering what happens when we apply our
- formula for 
+ We can show this explicitly by considering what happens when we apply our formula for 
 \begin_inset Formula $A^{k}$
 \end_inset
 
@@ -1019,16 +1112,18 @@ e^{At}\vec{j}_{i}=\sum_{k=0}^{\infty}\frac{A^{k}t^{k}}{k!}\vec{j}_{i}=\sum_{k=0}
 
 \end_inset
 
-In general, that's how we show the formula for 
+In general,
+ that's how we show the formula for 
 \begin_inset Formula $f(A)$
 \end_inset
 
- above: we Taylor expand each term, and the 
+ above:
+ we Taylor expand each term,
+ and the 
 \begin_inset Formula $A^{k}$
 \end_inset
 
- formula means that each term in the Taylor expansion has corresponding
- term multiplying 
+ formula means that each term in the Taylor expansion has corresponding term multiplying 
 \begin_inset Formula $\vec{j}_{i}$
 \end_inset
 
@@ -1048,25 +1143,26 @@ More than double roots
 \end_layout
 
 \begin_layout Standard
-In the rare case of two Jordan vectors from a triple root, you will have
- a Jordan vector 
+In the rare case of two Jordan vectors from a triple root,
+ you will have a Jordan vector 
 \begin_inset Formula $\vec{x}_{i}^{(3)}$
 \end_inset
 
  and get a 
-\begin_inset Formula $f(A)\vec{x}_{i}^{(3)}=f(\lambda)\vec{x}_{i}^{(3)}+f'(\lambda)\vec{j}_{i}+f''(\lambda)\vec{x}_{i}$
+\begin_inset Formula $f(A)\vec{x}_{i}^{(3)}=f(\lambda)\vec{x}_{i}^{(3)}+f'(\lambda)\vec{j}_{i}+\frac{f''(\lambda)}{2}\vec{x}_{i}$
 \end_inset
 
-, where the 
-\begin_inset Formula $f''$
+,
+ where the 
+\begin_inset Formula $\frac{f''}{2}$
 \end_inset
 
  term will give you 
-\begin_inset Formula $k(k-1)\lambda_{i}^{k-2}$
+\begin_inset Formula $\frac{k(k-1)}{2}\lambda_{i}^{k-2}$
 \end_inset
 
  and 
-\begin_inset Formula $t^{2}e^{\lambda_{i}t}$
+\begin_inset Formula $\frac{t^{2}}{2}e^{\lambda_{i}t}$
 \end_inset
 
  for 
@@ -1078,12 +1174,11 @@ In the rare case of two Jordan vectors from a triple root, you will have
 \end_inset
 
  respectively.
- A quadruple root with one eigenvector and three Jordan vectors will give
- you 
-\begin_inset Formula $f'''$
+ A quadruple root with one eigenvector and three Jordan vectors will give you 
+\begin_inset Formula $\frac{f'''}{3!}$
 \end_inset
 
- terms (that is, 
+ terms (hence 
 \begin_inset Formula $k^{3}$
 \end_inset
 
@@ -1091,9 +1186,12 @@ In the rare case of two Jordan vectors from a triple root, you will have
 \begin_inset Formula $t^{3}$
 \end_inset
 
- terms), and so on.
- The theory is quite pretty, but doesn't arise often in practice so I will
- skip it; it is straightforward to work it out on your own if you are interested.
+ terms),
+ and so on,
+ very much like a Taylor series.
+ The theory is quite pretty,
+ but doesn't arise often in practice so I will skip it;
+ it is straightforward to work it out on your own if you are interested.
 \end_layout
 
 \begin_layout Subsection
@@ -1112,7 +1210,8 @@ Let's try this for our example
 \end{array}\right)$
 \end_inset
 
- from above, which has an eigenvector 
+ from above,
+ which has an eigenvector 
 \begin_inset Formula $\vec{x}_{1}=(1,0)$
 \end_inset
 
@@ -1138,12 +1237,13 @@ Let's try this for our example
 \end_inset
 
 .
- As usual, our first step is to write 
+ As usual,
+ our first step is to write 
 \begin_inset Formula $\vec{u}_{0}$
 \end_inset
 
- in the basis of the eigenvectors...except that now we also include the generalized
- eigenvectors to get a complete basis: 
+ in the basis of the eigenvectors...except that now we also include the generalized eigenvectors to get a complete basis:
+ 
 \begin_inset Formula 
 \[
 \vec{u}_{0}=\left(\begin{array}{c}
@@ -1154,11 +1254,14 @@ Let's try this for our example
 
 \end_inset
 
-Now, computing 
+Now,
+ computing 
 \begin_inset Formula $A^{k}\vec{u}_{0}$
 \end_inset
 
- is easy, from our formula above: 
+ is easy,
+ from our formula above:
+ 
 \begin_inset Formula 
 \begin{eqnarray*}
 A^{k}\vec{u}_{0} & = & A^{k}\vec{x}_{1}+2A^{k}\vec{j}_{1}=\lambda_{1}^{k}\vec{x}_{1}+2\lambda_{1}^{k}\vec{j}_{1}+2k\lambda_{1}^{k-1}\vec{x}_{1}\\
@@ -1176,7 +1279,8 @@ A^{k}\vec{u}_{0} & = & A^{k}\vec{x}_{1}+2A^{k}\vec{j}_{1}=\lambda_{1}^{k}\vec{x}
 
 \end_inset
 
-For example, this is the solution to the recurrence 
+For example,
+ this is the solution to the recurrence 
 \begin_inset Formula $\vec{u}_{k+1}=A\vec{u}_{k}$
 \end_inset
 
@@ -1189,7 +1293,9 @@ For example, this is the solution to the recurrence
 \begin_inset Formula $|\lambda_{1}|=1\leq1$
 \end_inset
 
-, the solution still blows up, but it blows up 
+,
+ the solution still blows up,
+ but it blows up 
 \emph on
 linearly
 \emph default
@@ -1205,7 +1311,8 @@ Consider instead
 \begin_inset Formula $e^{At}\vec{u}_{0}$
 \end_inset
 
-, which is the solution to the system of ODEs 
+,
+ which is the solution to the system of ODEs 
 \begin_inset Formula $\frac{d\vec{u}(t)}{dt}=A\vec{u}(t)$
 \end_inset
 
@@ -1214,7 +1321,8 @@ Consider instead
 \end_inset
 
 .
- In this case, we get:
+ In this case,
+ we get:
 \begin_inset Formula 
 \begin{eqnarray*}
 e^{At}\vec{u}_{0} & = & e^{At}\vec{x}_{1}+2e^{At}\vec{j}_{1}=e^{\lambda_{1}t}\vec{x}_{1}+2e^{\lambda_{1}t}\vec{j}_{1}+2te^{\lambda_{1}t}\vec{x}_{1}\\
@@ -1232,11 +1340,13 @@ e^{At}\vec{u}_{0} & = & e^{At}\vec{x}_{1}+2e^{At}\vec{j}_{1}=e^{\lambda_{1}t}\ve
 
 \end_inset
 
-In this case, the solution blows up exponentially since 
+In this case,
+ the solution blows up exponentially since 
 \begin_inset Formula $\lambda_{1}=1>0$
 \end_inset
 
-, but we have an 
+,
+ but we have an 
 \emph on
 additional
 \emph default
@@ -1248,15 +1358,14 @@ t
 \end_layout
 
 \begin_layout Standard
-Those of you who have taken 18.03 are probably familiar with these terms
- multiplied by 
+Those of you who have taken 18.03 are probably familiar with these terms multiplied by 
 \begin_inset Formula $t$
 \end_inset
 
  in the case of a repeated root.
- In 18.03, it is presented simply as a guess for the solution that turns
- out to work, but here we see that it is part of a more general pattern
- of Jordan vectors for defective matrices.
+ In 18.03,
+ it is presented simply as a guess for the solution that turns out to work,
+ but here we see that it is part of a more general pattern of Jordan vectors for defective matrices.
 \end_layout
 
 \begin_layout Section
@@ -1268,11 +1377,13 @@ For a diagonalizable matrix
 \begin_inset Formula $A$
 \end_inset
 
-, we made a matrix 
+,
+ we made a matrix 
 \begin_inset Formula $S$
 \end_inset
 
- out of the eigenvectors, and saw that multiplying by 
+ out of the eigenvectors,
+ and saw that multiplying by 
 \begin_inset Formula $A$
 \end_inset
 
@@ -1284,7 +1395,8 @@ For a diagonalizable matrix
 \begin_inset Formula $\Lambda=S^{-1}AS$
 \end_inset
 
- is the diagonal matrix of eigenvalues, the 
+ is the diagonal matrix of eigenvalues,
+ the 
 \emph on
 diagonalization
 \emph default
@@ -1293,11 +1405,13 @@ diagonalization
 \end_inset
 
 .
- Equivalently, 
+ Equivalently,
+ 
 \begin_inset Formula $AS=\Lambda S$
 \end_inset
 
-: 
+:
+ 
 \begin_inset Formula $A$
 \end_inset
 
@@ -1306,11 +1420,14 @@ diagonalization
 \end_inset
 
  by the corresponding eigenvalue.
- Now, we will do exactly the same steps for a defective matrix 
+ Now,
+ we will do exactly the same steps for a defective matrix 
 \begin_inset Formula $A$
 \end_inset
 
-, using the basis of eigenvectors and Jordan vectors, and obtain the 
+,
+ using the basis of eigenvectors and Jordan vectors,
+ and obtain the 
 \series bold
 Jordan form
 \series default
@@ -1330,7 +1447,8 @@ Let's consider a simple case of a
 \begin_inset Formula $4\times4$
 \end_inset
 
- first, in which there is only 
+ first,
+ in which there is only 
 \emph on
 one
 \emph default
@@ -1346,7 +1464,8 @@ one
 \begin_inset Formula $\vec{j}_{2}$
 \end_inset
 
-, and the other two eigenvalues 
+,
+ and the other two eigenvalues 
 \begin_inset Formula $\lambda_{1}$
 \end_inset
 
@@ -1368,7 +1487,8 @@ one
 \end_inset
 
  from this basis of four vectors (3 eigenvectors and 1 Jordan vector).
- Now, consider what happends when we multiply 
+ Now,
+ consider what happends when we multiply 
 \begin_inset Formula $A$
 \end_inset
 
@@ -1376,7 +1496,8 @@ one
 \begin_inset Formula $M$
 \end_inset
 
-: 
+:
+ 
 \begin_inset Formula 
 \begin{eqnarray*}
 AM & = & (\lambda_{1}\vec{x}_{1},\lambda_{2}\vec{x}_{2},\lambda_{2}\vec{j}_{2}+\vec{x}_{2},\lambda_{3}\vec{x}_{3}).\\
@@ -1390,7 +1511,8 @@ AM & = & (\lambda_{1}\vec{x}_{1},\lambda_{2}\vec{x}_{2},\lambda_{2}\vec{j}_{2}+\
 
 \end_inset
 
-That is, 
+That is,
+ 
 \begin_inset Formula $A=MJM^{-1}$
 \end_inset
 
@@ -1402,14 +1524,15 @@ That is,
 \emph on
 almost
 \emph default
- diagonal: it has 
+ diagonal:
+ it has 
 \begin_inset Formula $\lambda's$
 \end_inset
 
- along the diagonal, but it 
+ along the diagonal,
+ but it 
 \emph on
-also has 1's above the diagonal for the columns corresponding to generalized
- eigenvectors
+also has 1's above the diagonal for the columns corresponding to generalized eigenvectors
 \emph default
 .
  This is exactly the Jordan form of the matrix 
@@ -1421,7 +1544,9 @@ also has 1's above the diagonal for the columns corresponding to generalized
 \begin_inset Formula $J$
 \end_inset
 
-, of course, has the same eigenvalues as 
+,
+ of course,
+ has the same eigenvalues as 
 \begin_inset Formula $A$
 \end_inset
 
@@ -1433,7 +1558,8 @@ also has 1's above the diagonal for the columns corresponding to generalized
 \begin_inset Formula $J$
 \end_inset
 
- are similar, but 
+ are similar,
+ but 
 \begin_inset Formula $J$
 \end_inset
 
@@ -1465,17 +1591,21 @@ Jordan block
 \end_layout
 
 \begin_layout Standard
-The generalization of this, when you perhaps have more than one repeated
- root, and perhaps the multiplicity of the root is greater than 2, is fairly
- obvious, and leads immediately to the formula given without proof in section
- 6.6 of the textbook.
- What I want to emphasize here, however, is not so much the formal theorem
- that a Jordan form exists, but how to 
+The generalization of this,
+ when you perhaps have more than one repeated root,
+ and perhaps the multiplicity of the root is greater than 2,
+ is fairly obvious,
+ and leads immediately to the formula given without proof in section 6.6 of the textbook.
+ What I want to emphasize here,
+ however,
+ is not so much the formal theorem that a Jordan form exists,
+ but how to 
 \emph on
 use
 \emph default
- it via the Jordan vectors: in particular, that generalized eigenvectors
- give us 
+ it via the Jordan vectors:
+ in particular,
+ that generalized eigenvectors give us 
 \emph on
 linearly growing
 \emph default
@@ -1495,17 +1625,20 @@ linearly growing
 \begin_inset Formula $e^{At}$
 \end_inset
 
-, respectively.
+,
+ respectively.
 \end_layout
 
 \begin_layout Standard
-Computationally, the Jordan form is famously problematic, because with any
- slight random perturbation to 
+Computationally,
+ the Jordan form is famously problematic,
+ because with any slight random perturbation to 
 \begin_inset Formula $A$
 \end_inset
 
  (e.g.
- roundoff errors) the matrix typically becomes diagonalizable, and the 
+ roundoff errors) the matrix typically becomes diagonalizable,
+ and the 
 \begin_inset Formula $2\times2$
 \end_inset
 
@@ -1518,7 +1651,8 @@ Computationally, the Jordan form is famously problematic, because with any
 \begin_inset Formula $X$
 \end_inset
 
- of eigenvectors, but it is 
+ of eigenvectors,
+ but it is 
 \emph on
 nearly singular
 \emph default
@@ -1530,17 +1664,19 @@ ill conditioned
 \begin_inset Quotes erd
 \end_inset
 
-): for a 
+):
+ for a 
 \series bold
-nearly defective matrix, the eigenvectors are 
+nearly defective matrix,
+ the eigenvectors are 
 \emph on
 almost
 \emph default
  linearly dependent
 \series default
 .
- This makes eigenvectors a problematic way of looking at nearly defective
- matrices as well, because they are so sensitive to errors.
+ This makes eigenvectors a problematic way of looking at nearly defective matrices as well,
+ because they are so sensitive to errors.
  Finding an 
 \emph on
 approximate
@@ -1553,8 +1689,10 @@ nearly
 \series bold
 Wilkinson problem
 \series default
- in numerical linear algebra, and has a number of tricky solutions.
- Alternatively, there are approaches like 
+ in numerical linear algebra,
+ and has a number of tricky solutions.
+ Alternatively,
+ there are approaches like 
 \begin_inset Quotes eld
 \end_inset
 
@@ -1562,8 +1700,8 @@ Schur factorization
 \begin_inset Quotes erd
 \end_inset
 
- or the SVD that lead to nice orthonormal bases for any matrix, but aren't
- nearly as simple to use as eigenvectors.
+ or the SVD that lead to nice orthonormal bases for any matrix,
+ but aren't nearly as simple to use as eigenvectors.
 \end_layout
 
 \end_body
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