From 3a07849b8dab10632aa2da45733d35b7621d14ee Mon Sep 17 00:00:00 2001 From: Robert Date: Wed, 24 Mar 2021 22:32:20 +0100 Subject: [PATCH] added eigenvalues --- chapters/sections/eigenvalue.tex | 301 ++++++++++++++++++++++++++++++- script.pdf | Bin 528340 -> 545793 bytes 2 files changed, 300 insertions(+), 1 deletion(-) diff --git a/chapters/sections/eigenvalue.tex b/chapters/sections/eigenvalue.tex index a615627..594df2b 100644 --- a/chapters/sections/eigenvalue.tex +++ b/chapters/sections/eigenvalue.tex @@ -13,6 +13,305 @@ \[ \set[Av = \lambda v]{v \in \field^n} =: E_{\lambda} \] - eigenspace belonging to $\lambda$ + eigenspace belonging to $\lambda$. \end{defi} + +\begin{eg} + Let + \[ + A = \begin{pmatrix} + 2 & 1 & -1 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 + \end{pmatrix} + \] + Then + \begin{align*} + A \cdot \begin{pmatrix} + 1 \\ 0 \\ 0 + \end{pmatrix} + &= \begin{pmatrix} + 2 \\ 0 \\ 0 + \end{pmatrix} + = 2 \cdot \begin{pmatrix} + 1 \\ 0 \\ 0 + \end{pmatrix} \\ + A \cdot \begin{pmatrix} + 1 \\ -1 \\ 0 + \end{pmatrix} + &= \begin{pmatrix} + 1 \\ -1 \\ 0 + \end{pmatrix} + = 1 \cdot \begin{pmatrix} + 1 \\ -1 \\ 0 + \end{pmatrix} \\ + A \cdot \begin{pmatrix} + 1 \\ 0 \\ 1 + \end{pmatrix} + &= \begin{pmatrix} + 1 \\ 0 \\ 1 + \end{pmatrix} + = 1 \cdot \begin{pmatrix} + 1 \\ 0 \\ 1 + \end{pmatrix} + \end{align*} + The eigenspaces are + \begin{align*} + E_2 &= \set[\kappa \in \realn]{\kappa \cdot \begin{pmatrix}1\\0\\0\end{pmatrix}} \\ + E_1 &= \set[\kappa, \rho \in \realn]{\kappa \cdot \begin{pmatrix}1\\-1\\0\end{pmatrix} + \rho \cdot \begin{pmatrix}1\\0\\1\end{pmatrix}} + = \spn\set{\begin{pmatrix}1\\-1\\0\end{pmatrix}, \begin{pmatrix}1\\0\\1\end{pmatrix}} + \end{align*} +\end{eg} + +\begin{rem} + The eigenspace to an eigenvalue $\lambda$ is a linear subspace. +\end{rem} + +\begin{rem} + We want to find $\lambda \in \field$, $v \in \field^n$ such that + \[ + Av = \lambda v \iff (\underbrace{A - \lambda I}_{\in \field^{n \times n}}) v = 0 + \] + If $(A - \lambda I)$ is invertible, then $v = 0$. So the interesting case is when $(A - \lambda I)$ not invertible. + \[ + (A - \lambda I) \text{ not invertible} \iff \det(A - \lambda I) = 0 + \] + This determinant is called the characteristic polynomial. This polynomial has degree $n$, and the eigenvalues are the roots of that polynomial. + So let $\lambda$ be an eigenvalue of $A$, then + \[ + (A - \lambda I) v = 0 + \] + is a linear equation system for the components of $v$. +\end{rem} + +\begin{eg} + Let + \[ + A = \begin{pmatrix} + 0 & 1 \\ -1 & 0 + \end{pmatrix} + \in \cmpln^{2 \times 2} + \] + The characteristic polynomial is + \[ + \det(A - \lambda I) = \begin{vmatrix} + -\lambda & 1 \\ -1 & -\lambda + \end{vmatrix} + = \lambda^2 + 1 + \] + Its roots are + \begin{multicols}{2} + \noindent + \[ \lambda_1 = i \] + \[ \lambda_2 = -i \] + \end{multicols} + \noindent To find the eigenvector belonging to $\lambda_1$, we declare $v_1 = (x, y) \in \cmpln^2$ and solve the linear equation system + \begin{multicols}{2}\noindent + \[ (A - \lambda_1 I) v_1 = 0 \] + \begin{align*} + -ix + 1y &= 0 \\ + -1x - iy &= 0 \\ + \end{align*} + \end{multicols} + \noindent It has the solutions $x = -i$ and $y = 1$, so + \[ + v_1 = \begin{pmatrix} + -i \\ 1 + \end{pmatrix} + \] + Doing the same for $v_2$ yields + \[ + v_2 = \begin{pmatrix} + i \\ 1 + \end{pmatrix} + \] + It is to be noted that the eigenvectors aren't unique (multiples of eigenvectors are also eigenvectors). +\end{eg} + +\begin{eg} + Let $D$ be a diagonal matrix, with the diagonal entries $\lambda_j$. Then + \[ + \det(D - \lambda I) = \begin{vmatrix} + \lambda_1 - \lambda & & & \\ + & \lambda_2 - \lambda & & \\ + & & \ddots & \\ + & & & \lambda_n - \lambda \\ + \end{vmatrix} + \] + The roots (eigenvalues) are $\lambda_1, \lambda_2, \cdots, \lambda_n$, and the eigenvectors are $De_i = \lambda_i e_i$. +\end{eg} + +\begin{defi} + $A \in \field^{n \times n}$ is called diagonalizable if there exists a basis of $\field^n$ that consists of eigenvectors. +\end{defi} + +\begin{thm} + A matrix $A \in \field^{n \times n}$ is diagonalizable, if and only if there exists a diagonal matrix $D$ and a invertible matrix $T$ such that + \[ + D = \inv{T}AT + \] +\end{thm} +\begin{proof} + Let $e_1, e_2, \cdots, e_n$ be the canonical basis of $\field^n$. Define $TD\inv{T} = A$, and let $\lambda_1, \cdots, \lambda_n$ be the diagonal entries of $D$. + Then we know that + \begin{equation} + De_i = \lambda_ie_i, ~~\forall i \in \set{1, \cdots n} + \end{equation} + Since $T$ is invertible, the $Te_1, \cdots Te_n$ form a basis. + \begin{equation} + A(Te_i) = T(\inv{T}AT)e_i = TDe_i = T\lambda_i e_i = \lambda_i (Te_i) + \end{equation} + Therefore $Te_i$ is an eigenvector of $A$ to the eigenvalue $\lambda_i$. Now let $v_1, \cdots, v_n$ be a basis of $\field^n$ and + \begin{equation} + Av_i = \lambda_iv_i, ~~\lambda_1, \cdots, \lambda_n \in \field^n + \end{equation} + Write write $v_1, \cdots, v_n$ as the columns of a matrix, therefore + \begin{subequations} + \begin{equation} + T = (v_1, v_2, \cdots, v_n) + \end{equation} + \begin{equation} + D = \begin{pmatrix} + \lambda_1 & & \\ + & \vdots & \\ + & & \lambda_n + \end{pmatrix} + \end{equation} + \end{subequations} + So $Te_i = v_i$, and thus + \begin{equation} + A(Te_i) = Av_i = \lambda_iv_i = \lambda_i(Te_i) = T\lambda_ie_i = TDe_i + \end{equation} + This means that $(AT - TD)e_i = 0$, $\forall i \in \set{1, \cdots, n}$. + \begin{equation} + \implies AT = TD + \end{equation} + $T$ is invertible (left as an exercise for the reader), and thus + \begin{equation} + \implies \inv{T}AT = D + \end{equation} +\end{proof} + +\begin{eg} + \begin{enumerate}[(i)] + \item Let + \[ + A = \begin{pmatrix} + 0 & 1 \\ -1 & 0 + \end{pmatrix} + \] + The eigenvalues and eigenvectors are + \begin{multicols}{2}\noindent + \[ + A \cdot \begin{pmatrix} + -i \\ 1 + \end{pmatrix} + = + i \begin{pmatrix} + -i \\ 1 + \end{pmatrix} + \] + \[ + A \cdot \begin{pmatrix} + i \\ 1 + \end{pmatrix} + = + -i \begin{pmatrix} + i \\ 1 + \end{pmatrix} + \] + \end{multicols} + \noindent Therefore + \[ + T = \begin{pmatrix} + -i & i \\ 1 & 1 + \end{pmatrix} + \] + which has the inverse + \[ + \inv{T} = \frac{1}{2} \begin{pmatrix} + i & 1 \\ -i & 1 + \end{pmatrix} + \] + Finally, + \[ + \inv{T}AT = \frac{1}{2} + \begin{pmatrix} + i & 1 \\ -i & 1 + \end{pmatrix} + \begin{pmatrix} + 1 & 1 \\ i & -i + \end{pmatrix} + = \frac{1}{2} + \begin{pmatrix} + 2i & 0 \\ 0 & -2i + \end{pmatrix} + = + \begin{pmatrix} + i & 0 \\ 0 & -i + \end{pmatrix} + \] + This is a diagonal matrix, therefore $A$ is diagonalizable. + + \item The matrix + \[ + \begin{pmatrix} + 0 & 1 \\ 0 & 0 + \end{pmatrix} + \] + is not diagonalizable since its only eigenvector is $(1, 0)^T$. + \end{enumerate} +\end{eg} + +\begin{rem} + For diagonal matrices the following is true + \[ + \begin{pmatrix} + \lambda_1 & & & \\ + & \lambda_2 & & \\ + & & \ddots & \\ + & & & \lambda_3 \\ + \end{pmatrix}^k + = + \begin{pmatrix} + \lambda_1^k & & & \\ + & \lambda_2^k & & \\ + & & \ddots & \\ + & & & \lambda_3^k \\ + \end{pmatrix} + \] + If $\inv{T}AT = D$ (where $D$ is a diagonal matrix), then + \[ + D^k = (\inv{T}AT)^k = \underbrace{\inv{T}AT \cdot \inv{T}AT \cdot \cdots}_{k \text{ times}} = \inv{T}A^kT + \] + \[ + \implies A^k = TD^k\inv{T} + \] +\end{rem} + +\begin{thm} + Let $A \in \realn^{n \times n}$ be a symmetric matrix, i.e. $A = A^T$. (Or if $A \in \cmpln^{n \times n}$ a self-adjoint matrix $A = A^H$). + Then $A$ has an orthonormal basis consisting of eigenvectors and is diagonalizable. +\end{thm} +\begin{proof} + Let $\lambda \in \cmpln$ be an eigenvalue of $A \in \field^{n \times n}$ with eigenvector $v \in \field^n$ and $A = A^H$. Then + \begin{equation} + \lambda\innerproduct{v}{v} = \innerproduct{v}{\lambda v} = \innerproduct{v}{Av} = \innerproduct{A^Hv}{v} = \innerproduct{Av}{v} = \innerproduct{\lambda v}{v} = \conj{\lambda} \innerproduct{v}{v} + \end{equation} + Therefore + \begin{equation} + (\lambda - \conj{\lambda})\underbrace{\innerproduct{v}{v}}_0 = 0 + \end{equation} + \begin{equation} + \implies (\lambda - \conj{\lambda}) = 0 \implies \lambda = \conj{\lambda} \implies \lambda \in \realn + \end{equation} + Now let $\lambda, \rho \in \realn$ be eigenvalues to the eigenvectors $v, w$, and require $\lambda \ne \rho$. Then + \begin{equation} + \rho\innerproduct{v}{w} = \innerproduct{v}{Aw} = \innerproduct{Av}{w} = \conj{\lambda}\innerproduct{v}{w} = \lambda\innerproduct{v}{w} + \end{equation} + And thus + \begin{equation} + \underbrace{(\rho - \lambda)}_{\ne 0} \underbrace{\innerproduct{v}{w}}_{=0} = 0 \implies v \perp w + \end{equation} +\end{proof} \end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index 6f8e0354be622b7c1f03fa1025010f23c3de81b5..388400de2da4301cca119d67c391f9f461a3e2a2 100644 GIT binary patch delta 98883 zcmaHRWmueB(k;-qySuwK4grF@yL%wGd*klb!CitSxVyVU2ofMbg1g%#XXcwb^PRc( zUu~({wcqaNskK+Fg*@`0a*DuE$WW*TKwu0cFy*We8J7;NCUXP8gQm5QilDu0 zHrP<07L;;;_OnoNvG1r^@<-~5-lroQW?!8OQhRC}*lOYN+Ro#BY1l@!4v(;&a#DU+R$s>&>Mbi{C^*#Y)e=RP=bz6M6+3cq{9L`$v)+iP?+pVTj# z`LU$9khc}|Noi;c=`=KkM?&x(__f*DE(+6^55jEDI1+H@Of33PqSbM0g`>`u&HC$@ znqpoDkV;WcGjdy#Vhr+~5Se~lyLu;yAG=q-yJOYSQwkyE3}L9sDHdowC;UC0&}{pt zvcwl&Uk>5Xgm?(=Jx1@5`loJ6K*&XkZA)7nwGJn0M_n6FXmTsHRHO{Pts{<&Vrv5C 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