From 774dbb16513cb6f720e8759fb5c719eeaa8e7cb9 Mon Sep 17 00:00:00 2001 From: Robert Date: Wed, 24 Mar 2021 21:02:15 +0100 Subject: [PATCH] started working on eigenvalue --- chapters/linear_algebra.tex | 2 + chapters/sections/eigenvalue.tex | 18 ++ chapters/sections/matrices.tex | 2 +- chapters/sections/scalar_product.tex | 272 +++++++++++++++++++++++++++ script.pdf | Bin 503518 -> 528340 bytes script.tex | 14 +- 6 files changed, 300 insertions(+), 8 deletions(-) create mode 100644 chapters/sections/eigenvalue.tex create mode 100644 chapters/sections/scalar_product.tex diff --git a/chapters/linear_algebra.tex b/chapters/linear_algebra.tex index 5858818..9787431 100644 --- a/chapters/linear_algebra.tex +++ b/chapters/linear_algebra.tex @@ -8,4 +8,6 @@ \subfile{sections/vector_spaces.tex} \subfile{sections/matrices.tex} \subfile{sections/determinant.tex} + \subfile{sections/scalar_product.tex} + \subfile{sections/eigenvalue.tex} \end{document} \ No newline at end of file diff --git a/chapters/sections/eigenvalue.tex b/chapters/sections/eigenvalue.tex new file mode 100644 index 0000000..a615627 --- /dev/null +++ b/chapters/sections/eigenvalue.tex @@ -0,0 +1,18 @@ +\documentclass[../../script.tex]{subfiles} +% !TEX root = ../../script.tex + +\begin{document} +\section{Eigenvalue problems} + +\begin{defi} + Let $A \in \field^{n \times n}$. Then $\lambda \in \field$ is called an eigenvalue of $A$, if + \[ + \exists v \in \field^n, ~v \ne 0: ~~Av = \lambda v + \] + Such a vector $v$ is called eigenvector. We call + \[ + \set[Av = \lambda v]{v \in \field^n} =: E_{\lambda} + \] + eigenspace belonging to $\lambda$ +\end{defi} +\end{document} \ No newline at end of file diff --git a/chapters/sections/matrices.tex b/chapters/sections/matrices.tex index 893b3b7..f1c37f3 100644 --- a/chapters/sections/matrices.tex +++ b/chapters/sections/matrices.tex @@ -60,7 +60,7 @@ $\field^{n\times m}$ is the space of all $n \times m$-matrices. The following op \item Conjugate transposition \[ - \conj{A} = \left(\overline{A}\right)^T + A^H = \left(\conj{A}\right)^T \] \end{enumerate} \end{defi} diff --git a/chapters/sections/scalar_product.tex b/chapters/sections/scalar_product.tex new file mode 100644 index 0000000..8cf69a4 --- /dev/null +++ b/chapters/sections/scalar_product.tex @@ -0,0 +1,272 @@ +\documentclass[../../script.tex]{subfiles} +% !TEX root = ../../script.tex + +\begin{document} +\section{Scalar Product} + +In this section $V$ will always denote a vector space and $\field$ a field (either $\realn$ or $\cmpln$). + +\begin{defi} + A scalar product is a mapping + \[ + \innerproduct{\cdot}{\cdot}: V \times V \longrightarrow \field + \] + that fulfils the following conditions: + $\forall v_1, v_2, w_1, w_2 \in V, ~~\lambda \in \field$ + \begin{align*} + \text{Linearity} && &\innerproduct{v_1}{w_1 + \lambda w_2} = \innerproduct{w_1}{w_1} + \lambda\innerproduct{v_1}{w_2} \\ + \text{Conjugated symmetry} && &\innerproduct{v_1}{w_1} = \conj{\innerproduct{w_1}{v_1}} \\ + \text{Positivity} && &\innerproduct{v_1}{v_1} \ge 0 \\ + \text{Definedness} && &\innerproduct{v_1}{v_2} = 0 \implies v_1 = 0 \\ + \text{Conjugated linearity} && &\innerproduct{v_1 + \lambda v_2}{w_1} = \innerproduct{v_1}{w_1} + \conj{\lambda} \innerproduct{v_2}{w_1} \\ + \end{align*} + The mapping + \begin{align*} + \norm{\cdot}: V &\longrightarrow \field \\ + v &\longmapsto \sqrt{\innerproduct{v}{v}} + \end{align*} +\end{defi} + +\begin{eg} + On $\realn^n$ the following is a scalar product + \[ + \innerproduct{(x_1, x_2, \cdots, x_n)^T}{(y_1, y_2, \cdots, y_n)^T} = \series[n]{k} x_ky_k + \] + The norm is then equivalent to the Pythagorean theorem + \[ + \norm{v} = \sqrt{\innerproduct{v}{v}} = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2} + \] + Analogously for $\cmpln^n$ + \[ + \innerproduct{(u_1, u_2, \cdots, u_n)^T}{(v_1, v_2, \cdots, v_n)^T} = \series[n]{k} \conj{u_k}v_k + \] +\end{eg} + +\begin{rem} +\begin{itemize} + \item The length of $v \in V$ is $\norm{v}$ + \item The distance between elements $v, w \in V$ is $\norm{v-w}$ + \item The angle $\phi$ between $v, w \in V$ is $\cos \phi = \frac{\innerproduct{v}{w}}{\norm{v}\cdot\norm{w}}$ +\end{itemize} +\end{rem} + +\begin{thm} + Let $v, w \in V$. Then + \begin{align*} + \text{Cauchy-Schwarz-Inequality} && &|\innerproduct{v}{w}| \le \norm{v}\norm{w} \\ + \text{Triangle Inequality} && &\norm{v + w} \le \norm{v} + \norm{w} + \end{align*} +\end{thm} +\begin{proof} + For $\lambda \in \field$ we know that + \begin{equation} + \begin{split} + 0 \le \innerproduct{v - \lambda w}{v - \lambda w} &= \innerproduct{v - \lambda w}{v} - \lambda\innerproduct{v - \lambda w}{w} \\ + &= \innerproduct{v}{v} - \conj{\lambda}\innerproduct{w}{v} - \lambda\innerproduct{v}{w} + \underbrace{\lambda\conj{\lambda}}_{|\lambda|^2}\innerproduct{w}{w} + \end{split} + \end{equation} + Let $\lambda = \frac{\innerproduct{w}{v}}{\norm{w}^2}$. Then + \begin{equation} + \begin{split} + 0 &\le \norm{v}^2 - \frac{\conj{\innerproduct{w}{v}}}{\norm{w}^2} \cdot \innerproduct{w}{v} - \frac{\innerproduct{w}{v}}{\norm{w}^2} \cdot \innerproduct{v}{w} + \frac{|\innerproduct{w}{v}|^2}{\norm{w}^4} \norm{w}^2 \\ + &= \norm{v}^2 - \frac{|\innerproduct{w}{v}|^2}{\norm{w}^2} - \cancel{\frac{|\innerproduct{w}{v}|^2}{\norm{w}^2}} + \cancel{\frac{|\innerproduct{w}{v}|^2}{\norm{w}^2}} \\ + &= \norm{v}^2 - \frac{|\innerproduct{w}{v}|^2}{\norm{w}^2} + \end{split} + \end{equation} + Through the monotony of the square root this implies that + \begin{equation} + |\innerproduct{w}{v}| \le \norm{v} \norm{w} + \end{equation} + To prove the triangle inequality, consider + \begin{equation} + \begin{split} + ||v + w||^2 &= \innerproduct{v+w}{v+w} \\ + &= \underbrace{\innerproduct{v}{v}}_{\norm{v}^2} + \innerproduct{v}{w} + \underbrace{\innerproduct{w}{v}}_{\conj{\innerproduct{v}{w}}} + \underbrace{\innerproduct{w}{w}}_{\norm{w}^2} \\ + &\le \norm{v}^2 + 2 \cdot \Re\innerproduct{v}{w} + \norm{w}^2 \\ + &\le \norm{v}^2 + 2\norm{v}\norm{w} + \norm{w}^2 \\ + &= (\norm{v} + \norm{w})^2 + \end{split} + \end{equation} + Using the same argument as above, this implies + \begin{equation} + \norm{v + w} \le \norm{v} + \norm{w} + \end{equation} +\end{proof} + +\begin{defi} + $v, w \in V$ are called orthogonal if + \[ + \innerproduct{v}{w} = 0 + \] + The elements $v_1, \cdots, v_m \in V$ are called an orthogonal set if they are non-zero and they are pairwise orthogonal. I.e. + \[ + \forall i,j \in \set{1, \cdots, m}: \innerproduct{v_i}{v_j} = 0 + \] + If $\norm{v_i} = 1$, then the $v_i$ are called an orthonormal set. If their span is $V$ they are an orthonormal basis. +\end{defi} + +\begin{thm} + If $v_1, \cdots, v_n$ are an orthonormal set, they are linearly independent. +\end{thm} +\begin{proof} + Let $\alpha_1, \cdots, \alpha_n \in \field$, such that + \begin{equation} + 0 = \alpha_1 v_1 + \alpha_2 v_2 + \cdots + \alpha_n v_n + \end{equation} + Then + \begin{equation} + \begin{split} + 0 &= \innerproduct{v_i}{0} = \innerproduct{v_i}{\alpha_1v_1 + \alpha_2v_2 + \cdots + \alpha_nv_n} \\ + &= \alpha_1\innerproduct{v_i}{v_1} + \alpha_2\innerproduct{v_i}{v_2} + \cdots + \alpha_n\innerproduct{v_i}{v_n} \\ + &= \alpha_i \innerproduct{v_i}{v_i} ~~i \in \set{1, \cdots, n} + \end{split} + \end{equation} + Since $v_i$ is not a zero vector, $\innerproduct{v_i}{v_i} \ne 0$, and thus $\alpha_i = 0$. Since $i$ is arbitrary, the $v_i$ are linearly independent. +\end{proof} + +\begin{eg} + \begin{enumerate}[(i)] + \item The canonical basis in $\realn^n$ is an orthonormal basis regarding the canonical scalar product. + \item Let $\phi \in \realn$. Then + \begin{align*} + v_1 = (\cos\phi, \sin\phi)^T && v_2 = (-\sin\phi, \cos\phi)^T + \end{align*} + are an orthonormal basis for $\realn^2$ + \end{enumerate} +\end{eg} + +\begin{thm} + Let $v_1, \cdots, v_n$ be an orthonormal basis of $V$. Then for $v \in V$: + \[ + v = \series[n]{i} \innerproduct{v_i}{v} v_i + \] +\end{thm} +\begin{proof} + Since $v_1, \cdots, v_n$ is a basis, + \begin{equation} + \exists \alpha_1, \cdots, \alpha_n \in \field: ~~v = \series[n]{i} \alpha_i v_i + \end{equation} + And therefore, for $j \in \set{1, \cdots, n}$ + \begin{equation} + \innerproduct{v_j}{v} = \series[n]{i} \alpha_i \innerproduct{v_j}{v_i} = \alpha_j \underbrace{\innerproduct{v_j}{v_j}}_{\norm{v_j}^2 = 1} + \end{equation} +\end{proof} + +\begin{thm} + Let $A \in \field^{m \times n}$ and $\innerproduct{\cdot}{\cdot}$ the canonical scalar product on $\field^n$. Then + \[ + \innerproduct{v}{Aw} = \innerproduct{A^Hv}{w} + \] +\end{thm} +\begin{proof} + First consider + \begin{multicols}{2} + \begin{subequations} + \noindent + \begin{equation} + (Aw)_i = \series[n]{j} A_{ij} w_i + \end{equation} + \begin{equation} + (A^H w)_j = \series[n]{i} A_{ji} v_i + \end{equation} + \end{subequations} + \end{multicols} + Now we can compute + \begin{equation} + \begin{split} + \innerproduct{v}{Aw} &= \series[n]{i} \conj{v_i} (Aw)_i = \series[n]{i}\left(\conj{v_i} \cdot \series[n]{j} A_{ij} w_j \right) = \series[n]{i}\series[n]{j} A_{ij} \conj{v_i}w_j \\ + &= \series[n]{j} \left(\series[n]{i} A{ij} \conj{v_i} \right) w_j = \series[n]{j} \left(\conj{\series[n]{i} \conj{A_{ij}} v_i}\right) w_j \\ + &= \series[n]{j} \conj{(A^H v)_j} \cdot w_j \\ + &= \innerproduct{A^H v}{w} + \end{split} + \end{equation} +\end{proof} + +\begin{defi} + A matrix $A \in \realn^{n \times n}$ is called orthogonal if + \[ + A^T A = AA^T = I + \] + or + \[ + A^T = A^{-1} + \] + The set of all orthogonal matrices + \[ + O(n) := \set[A^T A = I]{A \in \realn{n \times n}} + \] + is called the orthogonal group. + \[ + SO(n) = \set[A^TA = I \wedge \det A = 1]{A = \realn{n \times n}} \subset O(n) + \] + is called the special orthogonal group.6 +\end{defi} + +\begin{eg} + Let $\phi \in [0, 2\pi]$, then + \[ + A = \begin{pmatrix} + \cos \phi & -\sin \phi \\ + \sin \phi & \cos \phi + \end{pmatrix} + \] + is orthogonal. +\end{eg} + +\begin{rem} + \begin{enumerate}[(i)] + \item Let $A, B \in \field^{n \times n}$, then + \[ + AB = I \implies BA = I + \] + + \item \[ + 1 = \det I = \det A^TA = \det A^T \cdot \det A = {\det}^2 A + \] + + \item The $i$-$j$-component of $A^TA$ is equal to the canonical scalar product of the $i$-th row of $A^T$ and the $j$-th column of $A$. + Since the rows of $A^T$ are the columns of $A$, we can conclude that + \[ + A \text{ orthogonal} \iff \innerproduct{r_i}{r_j} = \delta_{ij} + \] + where the $r_i$ are the columns of $A$. In this case, the $r_i$ are an orthonormal basis on $\realn^n$. This works analogously for the rows. + + \item Let $A$ be orthogonal, and $x, y \in \realn^n$ + \begin{align*} + \innerproduct{Ax}{Ay} &= \innerproduct{A^TAx}{y} = \innerproduct{x}{y} \\ + \norm{Ax} &= \sqrt{\innerproduct{Ax}{Ax}} = \sqrt{\innerproduct{x}{x}} = \norm{x} + \end{align*} + $A$ perserves scalar products, lengths, distances and angles. These kinds of operations are called mirroring and rotation. + + \item Let $A, B \in O(n)$ + \[ + (AB)^T \cdot (AB) = B^TA^TAB = B^TIB = I + \] + This implies $(AB) \in O(n)$. It also implies $I \in O(n)$. Now consider $A \in O(n)$. Then + \[ + (\inv{A})^T \inv{A} = (A^T)^T \cdot A^T = AA^T = I + \] + This implies $\inv{A} \in O(T)$. Such a structure (a set with a multiplication operation, neutral element and multiplicative inverse) is called a group. + \end{enumerate} +\end{rem} + +\begin{eg} + $O(n)$, $SO(n)$, $\realn \setminus \set{0}$, $\cmpln \setminus \set{0}$, $Gl(n)$ (set of invertible matrices) and $S_n$ are all groups. +\end{eg} + +\begin{defi} + A matrix $U \in \cmpln^{n \times n}$ is called unitary if + \[ + U^H U = I = UU^H + \] + We also introduce + \[ + \set[U^HU = I]{U \in \cmpln{n \times n}} + \] + the unitary group, and + \[ + \set[U^HU = I \wedge \det U = 1]{U \in \cmpln{n \times n}} + \] + the special unitary group. +\end{defi} +\end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index 95f8ce6f69dc0d2eea87c7722502ec06953a2441..6f8e0354be622b7c1f03fa1025010f23c3de81b5 100644 GIT binary patch delta 191182 zcmb@vc|26_7dYO%V=epGMhMvnvlugl64|LN?K_o9dnu`=MY{?&o}`p^LPgSEO0+8N zDn+X$?I@+b=iV8aJAJ%geZRloA2{cEp8MQ;&-R?>obz1Is*LaLG&#~uiTG&UO`T_B 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