Geodesic
Matrix exponents and nilpotent algebras
Trudy Instituta matematiki, Tome 19 (2011) no. 2, pp. 37-46

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The differentiable at zero function $f$ acting in the matrix algebra $\mathrm M_n(\mathbb C)$ ($n\in\mathbb N$, $n>1$) with the properties $f(X+Y)=f(X)f(Y)$ and $f(0)=I$ is studied. The theorems about the general form of such functions are proved. 
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     author = {P. P. Zabreiko and A. N. Tanyhina},
     title = {Matrix exponents and nilpotent algebras},
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P. P. Zabreiko; A. N. Tanyhina. Matrix exponents and nilpotent algebras. Trudy Instituta matematiki, Tome 19 (2011) no. 2, pp. 37-46. http://geodesic.mathdoc.fr/item/TIMB_2011_19_2_a4/