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10.3: The Matrix Exponential as a Sum of Powers

  • Page ID
    21863
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    You may recall from Calculus that for any numbers aa and tt one may achieve eatea⁢t via

    \[e^{at} = \sum_{k=0}^{\infty} \frac{(at)^k}{k!} \nonumber\]

    The natural matrix definition is therefore

    \[e^{At} = \sum_{k=0}^{\infty} \frac{(At)^k}{k!} \nonumber\]

    where \(A^{0} = I\) is the n-by-n identity matrix.

    Example \(\PageIndex{1}\)

    The easiest case is the diagonal case, e.g.,

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

    for then

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

    and so

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

    Note that this is NOT the exponential of each element of \(A\).

    Example \(\PageIndex{2}\)

    As a second example let us suppose

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

    We recognize that its powers cycle, i.e.,

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

    \[A^3 = \begin{pmatrix} {0}&{-1}\\ {1}&{0} \end{pmatrix} \nonumber\]

    \[A^4 = \begin{pmatrix} {1}&{0}\\ {0}&{1} \end{pmatrix} \nonumber\]

    \[A^5 = \begin{pmatrix} {0}&{1}\\ {-1}&{0} \end{pmatrix} = A \nonumber\]

    and so

    \[e^{At} = \begin{pmatrix} {1-\frac{t^2}{2}+\frac{t^4}{4}+\cdots}&{t-\frac{t^3}{3!}+\frac{t^5}{5!}-\cdots}\\ {-t+\frac{t^3}{3!}-\frac{t^5}{5!}+\cdots}&{1-\frac{t^2}{2}+\frac{t^4}{4}+\cdots} \end{pmatrix} = \begin{pmatrix} {\cos(t)}&{\sin(t)}\\ {-\sin(t)}&{\cos(t)} \end{pmatrix} \nonumber\]

    Example \(\PageIndex{3}\)

    If

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

    then

    \[A^2 = A^3 = A^k = \begin{pmatrix} {0}&{1}\\ {0}&{0} \end{pmatrix} \nonumber\]

    and so

    \[e^{At} = (I+tA) \begin{pmatrix} {1}&{t}\\ {0}&{1} \end{pmatrix} \nonumber\]


    This page titled 10.3: The Matrix Exponential as a Sum of Powers 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; a detailed edit history is available upon request.