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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 at a t via

at=k=0atkk! a t k 0 a t k k (1)
The natural matrix definition is therefore
At=k=0Atkk! 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) At=t002t 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 At=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 At=I+tA=1t01 A t I t A 1 t 0 1

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