From 29fd154750c83e5c6c40c5435d3f97685612eebe Mon Sep 17 00:00:00 2001 From: Robert Date: Wed, 24 Mar 2021 13:46:04 +0100 Subject: [PATCH] Added matrix chapter --- chapters/sections/matrices.tex | 393 +++++++++++++++++++++++++++++++++ script.pdf | Bin 437507 -> 464096 bytes script.tex | 2 +- 3 files changed, 394 insertions(+), 1 deletion(-) diff --git a/chapters/sections/matrices.tex b/chapters/sections/matrices.tex index fd1ffb5..893b3b7 100644 --- a/chapters/sections/matrices.tex +++ b/chapters/sections/matrices.tex @@ -102,6 +102,399 @@ $\field^{n\times m}$ is the space of all $n \times m$-matrices. The following op \implies A(CE) = A(CE) \end{equation} \end{proof} + + \item Matrix multiplication is NOT commutative. First off, $AB$ and $BA$ are only well defined when $A \in \field^{n \times m}$ and $B \in \field^{m \times n}$. Example: + \[ + \begin{pmatrix} + 0 & 1 \\ 0 & 0 + \end{pmatrix} + \begin{pmatrix} + 0 & 0 \\ 1 & 0 + \end{pmatrix} + = + \begin{pmatrix} + 1 & 0 \\ 0 & 0 + \end{pmatrix} + \neq + \begin{pmatrix} + 0 & 0 \\ 1 & 0 + \end{pmatrix} + \begin{pmatrix} + 0 & 1 \\ 0 & 0 + \end{pmatrix} + = + \begin{pmatrix} + 0 & 0 \\ 0 & 1 + \end{pmatrix} + \] + + \item Let $n, m \in \natn$. There exists exactly one neutral additive element in $\field^{n \times m}$, which is the zero matrix. + Multiplication with the zero matrix yields a zero matrix. + + \item We define + \[ + \delta_{ij} = \begin{cases} + 1 ,& i = j \\ + 0 &\text{else} + \end{cases} + \] + The respective matrix $I = (\delta_{ij}) \in \field^{n \times m}$ is called the identity matrix. + + \item $A \ne 0$ and $B \ne 0$ can still result in $AB = 0$: + \[ + \begin{pmatrix} + 0 & 1 \\ 0 & 0 + \end{pmatrix}^2 + = + \begin{pmatrix} + 0 & 0 \\ 0 & 0 + \end{pmatrix} + \] \end{enumerate} \end{rem} + +\begin{eg}[Linear equation system] + Consider the following linear equation system + \begin{align*} + a_{11}x_1 + a_{12}x_2 + \cdots + a_{1m}x_m &= b_1 \\ + a_{21}x_1 + a_{22}x_2 + \cdots + a_{2m}x_m &= b_2 \\ + \vdots \\ + a_{n1}x_1 + a_{n2}x_2 + \cdots + a_{nm}x_m &= b_n \\ + \end{align*} + This can be rewritten using matrices + \begin{align*} + A = \begin{pmatrix} + a_{11} & \cdots & a_{1m} \\ + \vdots & \ddots & \vdots \\ + a_{n1} & \cdots & a_{nm} + \end{pmatrix} + && + x = \begin{pmatrix} + x_1 \\ \vdots \\ x_m + \end{pmatrix} + && + b = \begin{pmatrix} + b_1 \\ \vdots \\ b_n + \end{pmatrix} + \end{align*} + Which results in + \[ + Ax = B, ~~~A \in \field^{m \times n}, x \in \field^{m \times 1}, b \in \field^{n \times 1} + \] + Such an equation system is called homogeneous if $b = 0$. +\end{eg} + +\begin{thm} +Let $A \in \field^{n \times m}, b \in \field^n$. The solution set of the homogeneous equation system $Ax = 0$, (that means $\set[Ax = 0]{x \in \field^{m}} \subset \field^m$) is a linear subspace. +If $x$ and $\tilde{x}$ are solutions of the inhomogeneous system $Ax = b$, then $x-\tilde{x}$ solves the corresponding homogeneous problem. +\end{thm} +\begin{proof} + $A \cdot 0 = 0$ shows that $Ax = 0$ has a solution. Let $x, y$ be solutions, i.e. $Ax = 0$ and $Ay = 0$. Then $\forall \alpha \in \field$: + \begin{equation} + A(x + \alpha y) = Ax + A(\alpha y) = \underbrace{Ax}_0 + \alpha(\underbrace{Ay}_0) = 0 + \end{equation} + \begin{equation} + \implies x + \alpha y \in \set[Ax = 0]{x \in \field^m} + \end{equation} + + Next, let $x, \tilde{x}$ be solutions of $Ax = b$, i.e. + \begin{equation} + Ax = b, ~A\tilde{x} = b + \end{equation} + Then + \begin{equation} + A(x - \tilde{x}) = Ax - A\tilde{x} = b - b = 0 + \end{equation} + Therefore, $x - \tilde{x}$ is the solution of the homogeneous equation system +\end{proof} + +\begin{rem}[Finding all solutions] + First find a basis $e_1, \cdots, e_k$ of + \[ + \set[Ax = 0]{x \in \field^m} + \] + Next find some $x_0 \in \field^m$ such that $Ax_0 = b$. + Then every solution of $Ax = b$ can be written as + \[ + x = x_0 + \alpha_1 e_1 + \cdots + \alpha_k e_k + \] +\end{rem} + +\begin{eg} + Let + \begin{align*} + A = \begin{pmatrix} + 1 & 2 & 0 & 0 & 1 \\ + 0 & 0 & 1 & 0 & 0 \\ + 0 & 0 & 0 & 1 & -1 \\ + 0 & 0 & 0 & 0 & 0 + \end{pmatrix} + && + b = \begin{pmatrix} + 1 \\ 2 \\ 3 \\ 4 + \end{pmatrix} + && + c = \begin{pmatrix} + 3 \\ 2 \\ 1 \\ 0 + \end{pmatrix} + \end{align*} + Then $Ax = b$ has no solution, since the fourth row would state $0 = 4$. However, $Ax = c$ has the particular solution + \[ + x = \begin{pmatrix} + 3 \\ 0 \\ 2 \\ 1 \\ 0 + \end{pmatrix} + \] + If we consider the homogeneous problem $Ay = 0$, we can come up with the solution + \[ + y = \begin{pmatrix} + -2 \\ 1 \\ 0 \\ 0 \\ 0 + \end{pmatrix} + y_2 + + \begin{pmatrix} + -1 \\ 0 \\ 0 \\ 1 \\ 1 + \end{pmatrix} + y_5 + \] + and in turn find the set of solutions + \begin{align*} + \set[Ay = 0]{y \in \field^5} &= \spn\set{(-2, 1, 0, 0, 0)^T, (-1, 0, 0, 1, 1)^T} \\ + \set[Ax = c]{x \in \field^5} &= \set{(3, 0, 2, 1, 0)^T + \alpha(-2, 1, 0, 0, 0)^T + \beta(-1, 0, 0, 1, 1)^T} + \end{align*} +\end{eg} + +\begin{defi}[Row Echelon Form] + A zero row is a row in a matrix containing only zeros. The first element of a row that isn't zero is called the pivot. + + A matrix in row echelon form must meet the following conditions + \begin{enumerate}[(i)] + \item Every zero row is at the bottom + \item The pivot of a row is always strictly to the right of the pivot of the row above it + \end{enumerate} + + A matrix in reduced row echelon form must additionally meet the following conditions + \begin{enumerate}[(i)] + \item All pivots are $1$ + \item The pivot is the only non-zero element of its column + \end{enumerate} +\end{defi} + +\begin{rem} + Let $A \in \field^{n \times m}$ and $b \in \field^n$. If $A$ is in reduced row echelon form, then $Ax = b$ can be solved through trivial rearranging. +\end{rem} + +\begin{defi}[Matrix row operations] +Let $A$ be a matrix. Then the following are row operations +\begin{enumerate}[(i)] + \item Swapping of rows $i$ and $j$ + \item Addition of row $i$ to row $j$ + \item Multiplication of a row by $\lambda \ne 0$ + \item Addition of row $i$ multiplied by $lambda$ to row $j$ +\end{enumerate} +\end{defi} + +\begin{thm}[Gaussian Elimination] +Every matrix can be converted into reduced row echelon form in finitely many row operations. +\end{thm} +\begin{proof}[Heuristic Proof] + If $A$ is a zero matrix the proof is trivial. But if it isn't: + \begin{itemize} + \item Find the first column containing a non-zero element. + \begin{itemize} + \item Swap rows such that this element is in the first row + \end{itemize} + \item Multiply every other row with multiples of the first row, such that all other entries in that column disappear. + \item Repeat, but ignore the first row this time + \end{itemize} + At the end of this the matrix will be in reduced row echelon form. +\end{proof} + +\begin{defi} + $A \in \field^{n \times n}$ is called invertible if there exists a multiplicative inverse. I.e. + \[ + \exists B \in \field^{n \times n}: ~~AB = BA = I + \] + We denote the multiplicative inverse as $\inv{A}$ +\end{defi} + +\begin{rem} + We have seen matrices $A \ne 0$ such that $A^2 = 0$. Such a matrix is not invertible. +\end{rem} + +\begin{thm} + Let $A, B, C \in \field^{n \times n}$, $B$ invertible and $A = BC$. Then + \[ + A \text{ invertible} \iff C \text{ invertible} + \] + Especially, the product of invertible matrices is invertible. +\end{thm} +\begin{proof} + Without proof. +\end{proof} + +\begin{rem} + Matrix multiplication with $A$ from the left doesn't "mix" the columns of matrix $B$ +\end{rem} + +\begin{thm} + Let $A$ be a matrix, and let $\tilde{A}$ be the result of row operations applied to $A$. Then + \[ + \exists T \text{ invertible}: ~~\tilde{A} = TA + \] + We say: The left multiplication with $T$ applies the row operations. +\end{thm} +\begin{proof}[Heuristic proof] + You can find invertible matrices $T_1, \cdots, T_n$ that each apply one row operation. Then we can see that + \begin{equation} + \tilde{A} = \underbrace{T_n T_{n-1} \cdots T_1}_T A + \end{equation} + Since $T$ is the product of invertible matrices, it must itself be invertible. +\end{proof} + +\begin{cor} + Let $A \in \field^{n \times m}$, $b \in \field^n$, $T \in \field^{n \times m}$. Then $Ax = b$ and $TAx = Tb$ have the same solution sets. +\end{cor} +\begin{proof} + If $Ax = b$ it is trivial that + \begin{equation} + Ax = b \implies TAx = Tb + \end{equation} + If $TAx = Tb$, then + \begin{equation} + Ax = \inv{T}TAx = \inv{T}Tb = b + \end{equation} +\end{proof} + +\begin{lem} + Let $A \in field^{n \times m}$ be in row echelon form. Then + \[ + A \text{ invertible} \iff \text{ The last row is not a zero row} + \] + and + \[ + A \text{ invertible} \iff \text{ All diagonal entries are non-zero} + \] +\end{lem} +\begin{proof} + Let $A$ be invertible with a zero-row as its last row. Then + \begin{equation} + (0, \cdots, 0, 1) \cdot A = (0, \cdots, 0, 0) + \end{equation} + Multiplying with $\inv{A}$ from the right would result in a contradiction. Therefore the last row of $A$ can't be a zero row. + + Now let the diagonal entries of $A$ be non-zero. This means we can use row operations to transform $A$ into the identity matrix, i.e. + \begin{equation} + \exists T \text{ invertible}: ~~TA = I \implies A = \inv{T} + \end{equation} +\end{proof} + +\begin{cor} + Let $A \in \field^{n \times n}$. Then + \[ + A \text{ invertible} \iff \text{Every row echelon form has non-zero diagonal entries} + \] + and + \[ + A \text{ invertible} \iff \text{The reduced row echelon form is the identity matrix} + \] +\end{cor} +\begin{proof} + Every row echelon form of $A$ has the form $TA$ with $T$ an invertible matrix. Especially, $\exists S \text{ invertible}$ such that $SA$ is + in reduced row echelon form. Then + \begin{equation} + TA \text{ invertible} \iff A \text{ invertible} + \end{equation} +\end{proof} + +\begin{rem} + Let $A \in \field^{n \times n}$ be invertible, $B \in \field^{n \times m}$. Our goal is to compute $\inv{A}B$. First, write $(A \,\vert\, B)$. + Now apply row operations until we reach the form $(I \,\vert\, \tilde{B})$. Let $S$ be the matrix realising these operations, i.e. $SA = I$. + Then $\tilde{B} = SB = \inv{A}B$. If $B = I$ this can be used to compute $\inv{A}$. +\end{rem} + +\begin{eg} + Let + \[ + A = \begin{pmatrix} + 1 & 1 & 1 \\ + 0 & 1 & 1 \\ + 0 & 0 & 1 + \end{pmatrix} + \] + Rewrite this as + \[ + \left( + \begin{array}{@{}ccc|ccc@{}} + 1 & 1 & 1 & 1 & 0 & 0 \\ + 0 & 1 & 1 & 0 & 1 & 0 \\ + 0 & 0 & 1 & 0 & 0 & 1 + \end{array} + \right) + \] + Turn this into + \[ + \left( + \begin{array}{@{}ccc|ccc@{}} + 1 & 1 & 0 & 1 & 0 & -1 \\ + 0 & 1 & 0 & 0 & 1 & -1 \\ + 0 & 0 & 1 & 0 & 0 & 1 + \end{array} + \right) + \] + And finally + \[ + \left( + \begin{array}{@{}ccc|ccc@{}} + 1 & 0 & 0 & 1 & -1 & 0 \\ + 0 & 1 & 0 & 0 & 1 & -1 \\ + 0 & 0 & 1 & 0 & 0 & 1 + \end{array} + \right) + \] + The right part of the above matrix is $\inv{A}$. +\end{eg} + +\begin{defi} + Let $A \in \field^{n \times m}$ and let $z_1, \cdots, z_n \in \field^{1 \times m}$ be the rows of $A$. The row space of $A$ is defined as + \[ + \spn\set{z_1, \cdots, z_n} + \] + The dimension of the row space is the row rank of the matrix. Analogously this works for the column space and the column rank. + Later we will be able to show that row rank and column rank are always equal. They're therefore simply called rank of the matrix. +\end{defi} + +\begin{thm} + The row operations don't effect the row space. +\end{thm} +\begin{proof} + It is obvious that multiplication with $\lambda$ and swapping of rows don't change the row space. Furthermore it is clear that every linear combination of + $z_1 + z_2, z_2, \cdots, z_n$ is also a linear combination of $z_1, z_2, \cdots, z_n$, and vice versa. +\end{proof} + +\begin{thm} + Let $A$ be in row echelon form. Then the non-zero rows of the matrix are a basis of the row space of the matrix. +\end{thm} +\begin{proof} + Let $z_1, \cdots, z_k \in \field^{1 \times n}$ be the non-zero rows of $A$. They create the space $\spn\set{z_1, \cdots, z_n}$, + since $z_k, \cdots z_n$ are only zero rows. Analogously, + \begin{equation} + \alpha_1 z_1 + \alpha_2 z_2 + \cdots + \alpha_k z_k = 0 + \end{equation} + Let $j$ be the index of the column of the pivot of $z_1$. Then $z_2, \cdots, z_k$ have zero entries in the $j$-th column. Therefore + \begin{equation} + \alpha_1 \underbrace{z_{ij}}_{\ne 0} = 0 \implies \alpha_1 = 0 + \end{equation} + By inductivity, this holds for every row. +\end{proof} + +\begin{rem} + \begin{enumerate}[(i)] + \item To compute the rank of $A$, bring $A$ into row echelon form and count the non-zero rows. + + \item Let $v_1, \cdots, v_m \in \field^n$. To find a basis for + \[ + \spn\set{v_1, \cdots v_m} + \] + write $v_1, \cdots, v_m$ as rows of a matrix and bring it into row echelon form. + \end{enumerate} +\end{rem} \end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index e412dfd7d83ed4d8d266a38794151c3e61174b59..14020df6bc8a68e336fa6f5674fff8318a00e5a6 100644 GIT binary patch delta 144822 zcmaI8byS;O_Wn(RySuwKBxpi$EADQ^DHL~FAjOJPq!3(Mq{WL%arYuEPLbkP9E!cP z&&+&ho|*UeN7g#I_ul6|x2%=Vy7qMvGDo%ik)|>m9hna-l$3|g1&^0{-UY@9JG>$l zVf)gb;>fQ2Cd^AiXr`lnLS~~-H8+pd8}s(UYh%BrOYH^G3w8QLS2RqQw3{MlN4B~J zq_6$f0yD%v-%)>&U&H3Ed@{31@3htY$*Ywk`NxfCR^sIzcKLBQ{lanxFP=Ahof42x zsP|{09#t5N;M=CoVj&eyP}7cRXvR})$QXZixQ0ghQrj2PCr3+5ixcNoJ}_@$?gWBW zBWkpZ04xQf^upH>!BJb}UP_}16te~ufuYt$CUCQ#4%O^SYl^!eQ5fM-7=ZdH4B(TG z$PgF=k+%#GfGgy!*nWRQ2B1P$8y8U2)RDwj#Ea_g=HcK<*qS-0INSp%x_vCL=5$UZ z6Qg*i_kK1ltM6iD5bBq+{G*6};d8e77@c^52qOihqXJ(5LgQnC7GHu2W0)$RDNDt~ zlO*9uJNz81ImBB9B~*vZ?&mxT%(B0vWto^&xzA>zL*b7dWCixOJ0M~-noF2sY1UDN 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