matrix exponential

The exponential of a real valued square matrixMathworldPlanetmath A, denoted by eA, is defined as

eA = k=01k!Ak
= I+A+12A2+

Let us check that eA is a real valued square matrix. Suppose M is a real number such |Aij|<M for all entries Aij of A. Then |(A2)ij|<nM2 for all entries in A2, where n is the order of A. (Alternatively, one could argue using matrix normsMathworldPlanetmath: We have ||eA||e||A|| for the 2-norm, and hence the entries of eA are bounded by M=||eA||.) Thus, in general, we have |(Ak)i,j|<nkMk+1. Since k=0nkk!Mk+1 converges, we see that eA converges to real valued n×n matrix.

Example 1. Suppose A is nilpotent, i.e., Ar=0 for some natural numberMathworldPlanetmath r. Then

eA = I+A+12!A2++1(r-1)!Ar-1.

Example 2. If A is diagonalizable, i.e., of the form A=LDL-1, where D is a diagonal matrixMathworldPlanetmath, then

eA = k=01k!(LDL-1)k
= k=01k!LDkL-1
= LeDL-1.

Further, if D=diag{a1,,an}, then Dk=diag{a1k,,ank} whence

eA = Ldiag{ea1,,ean}L-1.

For diagonalizable matrix A, it follows that deteA=etrA. However, this formulaMathworldPlanetmathPlanetmath is, in fact, valid for all A.

Let A be a square n×n real valued matrix. Then the matrix exponentialMathworldPlanetmath satisfies the following properties

  1. 1.

    For the n×n zero matrixMathworldPlanetmath O, eO=I, where I is the n×n identity matrixMathworldPlanetmath.

  2. 2.

    If A=Ldiag{a1,,an}L-1 for an invertiblePlanetmathPlanetmathPlanetmath n×n matrix L, then

  3. 3.

    If A and B commute, then eA+B=eAeB.

  4. 4.

    The trace of A and the determinantMathworldPlanetmath of eA are related by the formula


    In effect, eA is always invertible. The inversePlanetmathPlanetmathPlanetmathPlanetmath is given by

  5. 5.

    If eA is a rotational matrix, then trA=0.

A relevant example on property 3.
We report an interesting example where the cited property is valid. In the field of complex numbersMathworldPlanetmathPlanetmath consider the complex matrix

C=A+iB, (1)

being C hermitian, i.e. C=C¯ (here ”” and overline ”-” stand for tranposition and conjugation, respectively) and orthogonalMathworldPlanetmathPlanetmath, i.e C-1=C . From (1),


Since C is orthogonal, from the complex equation CC=I (I is the identity matrix), we have


whence the imaginary part leads to the equation

BA+AB=0. (2)

But C is also hermitian, so that


therefore A=A is symmetricPlanetmathPlanetmathPlanetmathPlanetmath, and B=-B is skew-symmetric. From these and (2), BA=AB, and this implies that exp(A)exp(B)=exp(A+B). So that, the real and imaginary parts of an orthogonal and hermitian matrix verifies the property. Likewise, it is easy to show that if the complex matrix is symmetric and unitary, its real an imaginaryPlanetmathPlanetmath componentsPlanetmathPlanetmath also verify this property.

Title matrix exponential
Canonical name MatrixExponential
Date of creation 2013-03-22 13:33:27
Last modified on 2013-03-22 13:33:27
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 13
Author mathcam (2727)
Entry type Definition
Classification msc 15A15
Classification msc 15-00
Related topic ProofOfEquivalenceOfFormulasForExp