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The Matrix Exponential

Module by: Steven Cox

Summary: This module describes the general idea of matrix exponential.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

The matrix exponential is a powerful means for representing the solution to nn linear, constant coefficient, differential equations. The initial value problem for such a system may be written x ′t=Axt x t A x t x0=x0 x 0 x0 where AA is the n-by-n matrix of coefficients. By analogy to the 1-by-1 case we might expect

xt=ⅇAtu x t A t u (1)
to hold. Our expectations are granted if we properly define ⅇAt A t . Do you see why simply exponentiating each element of At will no suffice?

There are at least 4 distinct (but of course equivalent) approaches to properly defining ⅇAt A t . The first two are natural analogs of the single variable case while the latter two make use of heavier matrix algebra machinery.

Please visit each of these modules to see the definition and a number of examples.

For a concrete application of these methods to a real dynamical system, please visit the Mass-Spring-Damper module.

Regardless of the approach, the matrix exponential may be shown to obey the 3 lovely properties

  1. ddtⅇAt=AⅇAt=ⅇAtA t A t A A t A t A
  2. ⅇAt1+t2=ⅇAt1ⅇAt2 A t1 t2 A t1 A t2
  3. ⅇAt A t is nonsingular and ⅇAt-1=ⅇ-At A t A t

Let us confirm each of these on the suite of examples used in the submodules.

Example 1

If A=1002 A 1 0 0 2 then ⅇAt=ⅇt00ⅇ2t A t t 0 0 2 t

  1. ddtⅇAt=ⅇt002ⅇ2t=1002ⅇt00ⅇ2t t A t t 0 0 2 2 t 1 0 0 2 t 0 0 2 t
  2. ⅇt1+t200ⅇ2t1+2t2=ⅇt1ⅇt200ⅇ2t1ⅇ2t2=ⅇt100ⅇ2t1ⅇt200ⅇ2t2 t1 t2 0 0 2 t1 2 t2 t1 t2 0 0 2 t1 2 t2 t1 0 0 2 t1 t2 0 0 2 t2
  3. ⅇAt-1=ⅇ-t00ⅇ-2t=ⅇ-At A t t 0 0 2 t A t

Example 2

If A=01-10 A 0 1 -1 0 then ⅇAt=costsint-sintcost A t t t t t

  1. ddtⅇAt=-sintcost-cost-sint t A t t t t t and AⅇAt=-sintcost-cost-sint A A t t t t t
  2. You will recognize this statement as a basic trig identity cost1+t2sint1+t2-sint1+t2cost1+t2=cost1sint1-sint1cost1cost2sint2-sint2cost2 t1 t2 t1 t2 t1 t2 t1 t2 t1 t1 t1 t1 t2 t2 t2 t2
  3. Using the cofactor expansion we find ⅇAt-1=cost-sintsintcost=cos-t-sin-tsin-tcos-t=ⅇ-At A t t t t t t t t t A t

Example 3

If A=0100 A 0 1 0 0 then ⅇAt=1t01 A t 1 t 0 1

  1. ddtⅇAt=0100=AⅇAt t A t 0 1 0 0 A A t
  2. 1t1+t201=1t1011t201 1 t1 t2 0 1 1 t1 0 1 1 t2 0 1
  3. 1t01-1=1-t01=ⅇ-At 1 t 0 1 1 t 0 1 A t

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