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Math Help - Exponential of an Operator

  1. #1
    Newbie Assassin0071's Avatar
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    Exponential of an Operator

    In class we recently learned that for a linear operator T: V \rightarrow V and function g(t) = a_0 + a_1t + \dots + a_nt^n one can define the operator g(T) = a_0I + a_1T + \dots + a_nT^n (where I is the identity transformation). We also recently learned about the exponential of a matrix. My question is that for a linear operator T: V \rightarrow V can the operator e^T be defined? (For example, like how e^A is defined for a matrix A) (I tried searching for information on it but all I found was information on the exponential of a matrix). Thanks.
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    Re: Exponential of an Operator

    What is e here and how is e^A defined for a matrix A?
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    Re: Exponential of an Operator

    e here is the constant, e = 2.718... (the one used for the exponential function f(x) = e^x.

    And for a matrix A, e^A is defined as follows:

    e^A = \sum_{n=0}^\infty \frac{A^n}{n!}
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    Re: Exponential of an Operator

    Quote Originally Posted by Assassin0071 View Post
    And for a matrix A, e^A is defined as follows:

    e^A = \sum_{n=0}^\infty \frac{A^n}{n!}
    Thanks.

    Quote Originally Posted by Assassin0071 View Post
    My question is that for a linear operator T: V \rightarrow V can the operator e^T be defined? (For example, like how e^A is defined for a matrix A) (I tried searching for information on it but all I found was information on the exponential of a matrix).
    Are you asking about operators on infinite-dimensional vector spaces? Because on a finite-dimensional space, every linear operator is given by a matrix, so the exponentiation of an operator is determined by the exponentiation of its matrix.
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