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Math Help - Exponential of matrices

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    Exponential of matrices

    Let T be a square matrix and I be the corresponding Identity Matrix.

    Show that if \| T-I \| is sufficiently small, there exists a linear Operator S such that e^S = T.

    Hint: expand log(1+x) in a Taylor series.
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    Quote Originally Posted by EinStone View Post
    Let T be a square matrix and I be the corresponding Identity Matrix.

    Show that if \| T-I \| is sufficiently small, there exists a linear Operator S such that e^S = T.

    Hint: expand log(1+x) in a Taylor series.
    Let V = T-I, and suppose that \|V\|<1. Then define S = \log(I+V) = V - \frac{V^2}2 + \frac{V^3}3 -\frac{V^4}4 + \ldots. The series converges absolutely, because \sum\frac{\|V^n\|}n \leqslant\sum\frac{\|V\|^n}n and that series has radius of convergence 1. Then e^S = \exp\log(I+V) = I+V = T.
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