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The Matrix Exponential as a Limit of Powers

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

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

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

eat=limit  k1+atkk a t k 1 a t k k
(1)
The natural matrix definition is therefore
eAt=limit  kI+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=( 10 02 ) A 1 0 0 2 for then I+Atkk=( 1+tkk0 01+2tkk ) I A t k k 1 t k k 0 0 1 2 t k k and so (recalling Equation 1 above) eAt=( et0 0e2t ) 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 t21 )( 1t2 t21 )=( 1t24t t1t24 ) 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=( 1t23tt327 t+t3271t23 ) 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+1tt316 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+1t2t325+t53125 t+2t325t53125-2t25+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=( 01 00 ) A 0 1 0 0 then I+Atkk=( 1t 01 ) I A t k k 1 t 0 1 for each value of kk and so eAt=( 1t 01 ) A t 1 t 0 1

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