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10.1: Visión general

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    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.

    1. La Matriz Exponencial como Límite de Poderes
    2. La Matriz Exponencial como suma de Poderes
    3. La Matriz Exponencial a través de la Transformación Laplace
    4. 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

    1. \(\frac{d}{dt}(e^{At}) = Ae^{At} = e^{At}A\)
    2. \(e^{A(t_{1}+t_{2})} = e^{At_{1}}e^{At_{2}}\)
    3. \(e^{At}\)es no singular y\((e^{At})^{-1} = e^{-(At)}\)

    Confirmemos cada uno de estos en el conjunto de ejemplos utilizados en los submódulos.

    Ejemplo\(\PageIndex{1}\)

    Si

    \[A = \begin{pmatrix} {1}&{0}\\ {0}&{2} \end{pmatrix} \nonumber\]

    entonces

    \[e^{At} = \begin{pmatrix} {e^t}&{0}\\ {0}&{e^{2t}} \end{pmatrix} \nonumber\]

    1. \(\frac{d}{dt}(e^{At}) = \begin{pmatrix} {e^t}&{0}\\ {0}&{e^{2t}} \end{pmatrix} = \begin{pmatrix} {1}&{0}\\ {0}&{2} \end{pmatrix} \begin{pmatrix} {e^t}&{0}\\ {0}&{e^{2t}} \end{pmatrix}\)
    2. \(\begin{pmatrix} {e^{t_{1}+t_{2}}}&{0}\\ {0}&{e^{2t_{1}+2t_{2}}} \end{pmatrix} = \begin{pmatrix} {e^{t_{1}}e^{t_{2}}}&{0}\\ {0}&{e^{2t_{1}}e^{2t_{2}}} \end{pmatrix} = \begin{pmatrix} {e^{t_{1}}}&{0}\\ {0}&{e^{2t_{1}}} \end{pmatrix} \begin{pmatrix} {e^{t_{2}}}&{0}\\ {0}&{e^{2t_{2}}} \end{pmatrix}\)
    3. \((e^{At})^{-1} = \begin{pmatrix} {e^{-t}}&{0}\\ {0}&{e^{-(2t)}} \end{pmatrix} = e^{-(At)}\)
    Ejemplo\(\PageIndex{2}\)

    Si

    \[A = \begin{pmatrix} {0}&{1}\\ {-1}&{0} \end{pmatrix} \nonumber\]

    entonces

    \[e^{At} = \begin{pmatrix} {\cos(t)}&{\sin(t)}\\ {-\sin(t)}&{\cos(t)} \end{pmatrix} \nonumber\]

    1. \(\frac{d}{dt}(e^{At}) = \begin{pmatrix} {-\sin(t)}&{\cos(t)}\\ {-\cos(t)}&{-\sin(t)} \end{pmatrix}\)y\(Ae^{At} = \begin{pmatrix} {-\sin(t)}&{\cos(t)}\\ {-\cos(t)}&{-\sin(t)} \end{pmatrix}\)
    2. Reconocerás esta declaración como una identidad trigonómica básica\(\begin{pmatrix} {\cos(t_{1}+t_{2})}&{\sin(t_{1}+t_{2})}\\ {-\sin(t_{1}+t_{2})}&{\cos(t_{1}+t_{2})} \end{pmatrix} = \begin{pmatrix} {\cos(t_{1})}&{\sin(t_{1})}\\ {-\sin(t_{1})}&{\cos(t_{1})} \end{pmatrix} \begin{pmatrix} {\cos(t_{2})}&{\sin(t_{2})}\\ {-\sin(t_{2})}&{\cos(t_{2})} \end{pmatrix}\)
    3. \((e^{At})^{-1} = \begin{pmatrix} {\cos(t)}&{-\sin(t)}\\ {\sin(t)}&{\cos(t)} \end{pmatrix} = \begin{pmatrix} {\cos(-t)}&{-\sin(-t)}\\ {\sin(-t)}&{\cos(-t)} \end{pmatrix} = e^{-(At)}\)
    Ejemplo\(\PageIndex{3}\)

    Si

    \[A = \begin{pmatrix} {0}&{1}\\ {0}&{0} \end{pmatrix} \nonumber\]

    entonces

    \[e^{At} = \begin{pmatrix} {1}&{t}\\ {0}&{1} \end{pmatrix} \nonumber\]

    1. \(\frac{d}{dt}(e^{At}) = \begin{pmatrix} {0}&{1}\\ {0}&{0} \end{pmatrix} = Ae^{At}\)
    2. \(\begin{pmatrix} {1}&{t_{1}+t_{2}}\\ {0}&{1} \end{pmatrix} = \begin{pmatrix} {1}&{t_{1}}\\ {0}&{1} \end{pmatrix} \begin{pmatrix} {1}&{t_{2}}\\ {0}&{1} \end{pmatrix}\)
    3. \(\begin{pmatrix} {1}&{t}\\ {0}&{1} \end{pmatrix}^{-1} = \begin{pmatrix} {1}&{-t}\\ {0}&{1} \end{pmatrix} = e^{-At}\)

    This page titled 10.1: Visión general is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by Steve Cox via source content that was edited to the style and standards of the LibreTexts platform.