the matrix exponential is well-defined

[[concept]]

Topics

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Corollary

For all $A\in M_n$ define $e^A={\sum }_{i=0}^{\infty }\frac{1}{i!}{A}^i$. This is well-defined. (see matrix exponential)

$||\cdot ||$ be any matrix norm. Then ${\sum }_{i=0}^{\infty }\left|\left|\frac{1}{i!}{A}^i\right|\right|\le {\sum }_{i=0}^{\infty }\frac{1}{i!}||A|{|}^i=e^{||A||}$

Let

by homogeneity and submultiplicity

When $i=0$ : can I say that $||A^0||\le ||A|{|}^0=1$ ? Well, no. But it still converges. (as long as $n$ is finite). I can also choose an induced matrix norm and using norm equivalence it is ok :)

References

References

See Also

Mentions

Mentions

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