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The Matrix Exponential as a Sum of Powers

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

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 ea⁢t a t via

ea⁢t=∑k=0∞a⁢tkk! a t k 0 a t k k
(1)
The natural matrix definition is therefore
eA⁢t=∑k=0∞A⁢tkk! 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=( 10 02 ) A 1 0 0 2 for then A⁢tk=( tk0 02⁢tk ) A t k t k 0 0 2 t k and so (recalling Equation 1 above) eA⁢t=( et0 0e2⁢t ) 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=( -10 0-1 ) A 2 -1 0 0 -1 A3=( 0-1 10 ) A 3 0 -1 1 0 A4=( 10 01 ) A 4 1 0 0 1 A5=( 01 -10 )=A A 5 0 1 -1 0 A and so eA⁢t=( 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=( 01 00 ) A 0 1 0 0 then A2=A3=Ak=( 01 00 ) A 2 A 3 A k 0 1 0 0 and so eA⁢t=I+t⁢A=( 1t 01 ) A t I t A 1 t 0 1

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