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The Matrix Exponential as a Sum of Powers

Module by: Steven Cox

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

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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