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Definition: The exponential of an nxn real or complex matrix A is the matrix obtained by substituting A for x and I for 1 into the Taylor's series for e^x, which is e^x= 1 + x/1! + (x^2)/2! + (x^3)/3! + ... Thus by definition, e^A = I + A/1! + (A^2)/2! + (A^3)/3! + ... or more generally given the variable scalar t, we can substitute tA for A and obtain: e^(tA) = I + tA/1! + ((tA)^2)/2! + ((tA)^3)/3! + ...
Source: Algebra, second edition by Michael Artin