10.1: Visión general
- Page ID
- 113052
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)La matriz exponencial es un medio poderoso para representar la solución a nn ecuaciones lineales, de coeficiente constante y diferenciales. El problema de valor inicial para un sistema de este tipo puede ser escrito
\[x′(t) = Ax(t) \nonumber\]
\[x(0) = x_{0} \nonumber\]
donde\(A\) es la matriz n-por-n de coeficientes. Por analogía con el caso 1 por 1 que podríamos esperar
\[x(t) = e^{At}u \nonumber\]
para sostener. Nuestras expectativas son otorgadas si definimos adecuadamente\(e^{At}\). ¿Ves por qué simplemente exponenciar cada elemento de no\(At\) será suficiente?
Hay al menos 4 enfoques distintos (pero por supuesto equivalentes) para definir adecuadamente\(e^{At}\). Los dos primeros son análogos naturales del caso de una sola variable mientras que los dos últimos hacen uso de maquinaria de álgebra matricial más pesada.
- La Matriz Exponencial como Límite de Poderes
- La Matriz Exponencial como suma de Poderes
- La Matriz Exponencial a través de la Transformación Laplace
- La matriz exponencial a través de valores propios y vectores propios
Visite cada uno de estos módulos para ver la definición y una serie de ejemplos.
Para una aplicación concreta de estos métodos a un sistema dinámico real, visite el módulo Mass Spring-Damper-module.
Independientemente del enfoque, se puede demostrar que la matriz exponencial obedece a las 3 propiedades encantadoras
- \(\frac{d}{dt}(e^{At}) = Ae^{At} = e^{At}A\)
- \(e^{A(t_{1}+t_{2})} = e^{At_{1}}e^{At_{2}}\)
- \(e^{At}\)es no singular y\((e^{At})^{-1} = e^{-(At)}\)
Confirmemos cada uno de estos en el conjunto de ejemplos utilizados en los submódulos.
Si
\[A = \begin{pmatrix} {1}&{0}\\ {0}&{2} \end{pmatrix} \nonumber\]
entonces
\[e^{At} = \begin{pmatrix} {e^t}&{0}\\ {0}&{e^{2t}} \end{pmatrix} \nonumber\]
- \(\frac{d}{dt}(e^{At}) = \begin{pmatrix} {e^t}&{0}\\ {0}&{e^{2t}} \end{pmatrix} = \begin{pmatrix} {1}&{0}\\ {0}&{2} \end{pmatrix} \begin{pmatrix} {e^t}&{0}\\ {0}&{e^{2t}} \end{pmatrix}\)
- \(\begin{pmatrix} {e^{t_{1}+t_{2}}}&{0}\\ {0}&{e^{2t_{1}+2t_{2}}} \end{pmatrix} = \begin{pmatrix} {e^{t_{1}}e^{t_{2}}}&{0}\\ {0}&{e^{2t_{1}}e^{2t_{2}}} \end{pmatrix} = \begin{pmatrix} {e^{t_{1}}}&{0}\\ {0}&{e^{2t_{1}}} \end{pmatrix} \begin{pmatrix} {e^{t_{2}}}&{0}\\ {0}&{e^{2t_{2}}} \end{pmatrix}\)
- \((e^{At})^{-1} = \begin{pmatrix} {e^{-t}}&{0}\\ {0}&{e^{-(2t)}} \end{pmatrix} = e^{-(At)}\)
Si
\[A = \begin{pmatrix} {0}&{1}\\ {-1}&{0} \end{pmatrix} \nonumber\]
entonces
\[e^{At} = \begin{pmatrix} {\cos(t)}&{\sin(t)}\\ {-\sin(t)}&{\cos(t)} \end{pmatrix} \nonumber\]
- \(\frac{d}{dt}(e^{At}) = \begin{pmatrix} {-\sin(t)}&{\cos(t)}\\ {-\cos(t)}&{-\sin(t)} \end{pmatrix}\)y\(Ae^{At} = \begin{pmatrix} {-\sin(t)}&{\cos(t)}\\ {-\cos(t)}&{-\sin(t)} \end{pmatrix}\)
- Reconocerás esta declaración como una identidad trigonómica básica\(\begin{pmatrix} {\cos(t_{1}+t_{2})}&{\sin(t_{1}+t_{2})}\\ {-\sin(t_{1}+t_{2})}&{\cos(t_{1}+t_{2})} \end{pmatrix} = \begin{pmatrix} {\cos(t_{1})}&{\sin(t_{1})}\\ {-\sin(t_{1})}&{\cos(t_{1})} \end{pmatrix} \begin{pmatrix} {\cos(t_{2})}&{\sin(t_{2})}\\ {-\sin(t_{2})}&{\cos(t_{2})} \end{pmatrix}\)
- \((e^{At})^{-1} = \begin{pmatrix} {\cos(t)}&{-\sin(t)}\\ {\sin(t)}&{\cos(t)} \end{pmatrix} = \begin{pmatrix} {\cos(-t)}&{-\sin(-t)}\\ {\sin(-t)}&{\cos(-t)} \end{pmatrix} = e^{-(At)}\)
Si
\[A = \begin{pmatrix} {0}&{1}\\ {0}&{0} \end{pmatrix} \nonumber\]
entonces
\[e^{At} = \begin{pmatrix} {1}&{t}\\ {0}&{1} \end{pmatrix} \nonumber\]
- \(\frac{d}{dt}(e^{At}) = \begin{pmatrix} {0}&{1}\\ {0}&{0} \end{pmatrix} = Ae^{At}\)
- \(\begin{pmatrix} {1}&{t_{1}+t_{2}}\\ {0}&{1} \end{pmatrix} = \begin{pmatrix} {1}&{t_{1}}\\ {0}&{1} \end{pmatrix} \begin{pmatrix} {1}&{t_{2}}\\ {0}&{1} \end{pmatrix}\)
- \(\begin{pmatrix} {1}&{t}\\ {0}&{1} \end{pmatrix}^{-1} = \begin{pmatrix} {1}&{-t}\\ {0}&{1} \end{pmatrix} = e^{-At}\)