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The Matrix Exponential as a Sum of Powers

Module by: Steven J. Cox. E-mail the author

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Summary: A formal description of the matrix exponential. The definition is given as well as examples of calculating it using Taylor's series.

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You may recall from Calculus that for any numbers aa and tt one may achieve ⅇat a t via

ⅇat=∑k=0∞atkk! a t k 0 a t k k (1)
The natural matrix definition is therefore
ⅇAt=∑k=0∞Atkk! A t k 0 A t k k (2)
where A0=I A 0 I is the nn-by-nn identity matrix.

Example 1

The easiest case is the diagonal case, e.g., A=1002 A 1 0 0 2 for then Atk=tk002tk A t k t k 0 0 2 t k and so (recalling Equation 1 above) ⅇAt=ⅇt00ⅇ2t A t t 0 0 2 t Note that this is NOT the exponential of each element of AA.

Example 2

As a second example let us suppose A=01-10 A 0 1 -1 0 We recognize that its powers cycle, i.e., A2=-100-1 A 2 -1 0 0 -1 A3=0-110 A 3 0 -1 1 0 A4=1001 A 4 1 0 0 1 A5=01-10=A A 5 0 1 -1 0 A and so ⅇAt=1−t22!+t44!−…t−t33!+t55!−…-t+t33!−t55!+…1−t22!+t44!−…=costsint-sintcost A t 1 t 2 2 t 4 4 … t t 3 3 t 5 5 … t t 3 3 t 5 5 … 1 t 2 2 t 4 4 … t t t t

Example 3

If A=0100 A 0 1 0 0 then A2=A3=Ak=0100 A 2 A 3 A k 0 1 0 0 and so ⅇAt=I+tA=1t01 A t I t A 1 t 0 1

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