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The Matrix Exponential as a Limit of Powers

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

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Summary: This module describes how to compute the matrix exponential using a limit of powers.

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

ⅇat=limk→∞1+atkk a t k 1 a t k k (1)
The natural matrix definition is therefore
ⅇAt=limk→∞I+Atkk A t k I A t k k (2)
where II is the n-by-n identity matrix.

Example 1

The easiest case is the diagonal case, e.g., A=1002 A 1 0 0 2 for then I+Atkk=1+tkk001+2tkk I A t k k 1 t k k 0 0 1 2 t k 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 concrete example let us suppose A=01-10 A 0 1 -1 0 From I+At=1t-t1 I A t 1 t t 1 I+At22=1t2-t211t2-t21=1−t24t-t1−t24 I A t 2 2 1 t 2 t 2 1 1 t 2 t 2 1 1 t 2 4 t t 1 t 2 4 I+At33=1−t23t−t327-t+t3271−t23 I A t 3 3 1 t 2 3 t t 3 27 t t 3 27 1 t 2 3 I+At44=-3t28+t4256+1t−t316-t+t316-3t28+t4256+1 I A t 4 4 -3 t 2 8 t 4 256 1 t t 3 16 t t 3 16 -3 t 2 8 t 4 256 1 I+At55=-2t25+t4125+1t−2t325+t53125-t+2t325−t53125-2t25+t4125+1 I A t 5 5 -2 t 2 5 t 4 125 1 t 2 t 3 25 t 5 3125 t 2 t 3 25 t 5 3125 -2 t 2 5 t 4 125 1 We discern a pattern: the diagonal elements are equal even polynomials while the off diagonal elements are equal but opposite odd polynomials. The degree of the polynomial will grow with kk and in the limit we 'recognize' ⅇAt=costsint-sintcost A t t t t t

Example 3

If A=0100 A 0 1 0 0 then I+Atkk=1t01 I A t k k 1 t 0 1 for each value of kk and so ⅇAt=1t01 A t 1 t 0 1

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