From fe578ad2c786f945c89c435dddda497de0073dd2 Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Sat, 14 Sep 2024 19:35:59 -0400 Subject: [PATCH] fix for section 3.1 --- notes/jordan-vectors.lyx | 524 +++++++++++++++++++++++++-------------- notes/jordan-vectors.pdf | Bin 111618 -> 121256 bytes 2 files changed, 331 insertions(+), 193 deletions(-) diff --git a/notes/jordan-vectors.lyx b/notes/jordan-vectors.lyx index 8965f0fc..50858a1f 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 \use_default_options false -\maintain_unincluded_children false +\maintain_unincluded_children no \language english \language_package default -\inputencoding auto -\fontencoding global +\inputencoding auto-legacy +\fontencoding auto \font_roman "times" "default" \font_sans "default" "default" \font_typewriter "default" "default" @@ -23,9 +23,13 @@ \font_default_family default \use_non_tex_fonts false \font_sc false -\font_osf false +\font_roman_osf false +\font_sans_osf false +\font_typewriter_osf false \font_sf_scale 100 100 \font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures false \graphics default \default_output_format default \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 \tracking_changes false \output_changes false +\change_bars false +\postpone_fragile_content false \html_math_output 0 \html_css_as_file 0 \html_be_strict false +\docbook_table_output 0 +\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 diff --git a/notes/jordan-vectors.pdf b/notes/jordan-vectors.pdf index 1bc2aca36e1978fa4cb4e7707d3c07fdcb367cf3..98c22c36c77c693674b081652e11f961fc2f7f09 100644 GIT binary patch delta 50032 zcmV()K;OTD=mw~>2aqHJGc}VD$0&cjTUn3fwh?~!uNc1s8hBgbA(GM>hOthZ*hUN( 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