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Matrix Exponential

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The power series that defines the exponential map e^x also defines a map between matrices. In particular,

exp(A)=e^(A)
(1)
=sum_(n=0)^(infty)(A^n)/(n!)
(2)
=I+A+(AA)/(2!)+(AAA)/(3!)+...,
(3)

converges for any square matrix A, where I is the identity matrix. The matrix exponential is implemented in the Wolfram Language as MatrixExp[m].

The Kronecker sum satisfies the nice property

exp(A) tensor exp(B)=exp(A direct sum B)
(4)

(Horn and Johnson 1994, p. 208).

Matrix exponentials are important in the solution of systems of ordinary differential equations (e.g., Bellman 1970).

In some cases, it is a simple matter to express the matrix exponential. For example, when A is a diagonal matrix, exponentiation can be performed simply by exponentiating each of the diagonal elements. For example, given a diagonal matrix

A=[a_1 0 ... 0; 0 a_2 ... 0; | | ... |; 0 0 ... a_k],
(5)

The matrix exponential is given by

exp(A)=[e^(a_1) 0 ... 0; 0 e^(a_2) ... 0; | | ... |; 0 0 ... e^(a_k)].
(6)

Since most matrices are diagonalizable, it is easiest to diagonalize the matrix before exponentiating it.

When A is a nilpotent matrix, the exponential is given by a matrix polynomial because some power of A vanishes. For example, when

A=[0 x z; 0 0 y; 0 0 0],
(7)

then

exp(A)=[1 x z+1/2xy; 0 1 y; 0 0 1]
(8)

and A^3=0.

For the zero matrix A=0,

e^(0)=I,
(9)

i.e., the identity matrix. In general,

e^(A)e^(-A)=e^(0)=I,
(10)

so the exponential of a matrix is always invertible, with inverse the exponential of the negative of the matrix. However, in general, the formula

e^(A)e^(B)=e^(A+B)
(11)

holds only when A and B commute, i.e.,

[A,B]=AB-BA=0.
(12)

For example,

exp([0 -x; 0 0]+[0 0; x 0])=[cosx -sinx; sinx cosx],
(13)

while

exp([0 -x; 0 0])exp([0 0; x 0])=[1 -x; 0 1][1 0; x 1] =[1-x^2 -x; x 1].
(14)

Even for a general 2×2 real matrix, however, the matrix exponential can be quite complicated

exp([a b; c d])=1/Delta[m_(11) m_(12); m_(21) m_(22)]
(15)

where

m_(11)=e^((a+d)/2)[Deltacosh(1/2Delta)+(a-d)sinh(1/2Delta)]
(16)
m_(12)=2be^((a+d)/2)sinh(1/2Delta)
(17)
m_(21)=2ce^((a+d)/2)sinh(1/2Delta)
(18)
m_(22)=e^((a+d)/2)[Deltacosh(1/2Delta)+(d-a)sinh(1/2Delta)],
(19)

and

Delta=sqrt((a-d)^2+4bc).
(20)

As Delta->0, this becomes

exp([a b; c d])=e^((a+d)/2)[1+1/2(a-d) b; c 1-1/2(a-d)].
(21)

See also

Exponential Function, Exponential Map, Kronecker Sum, Matrix, Matrix Power

Portions of this entry contributed by Todd Rowland

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References

Bellman, R. E. Introduction to Matrix Analysis, 2nd ed. New York: McGraw-Hill, 1970.Horn, R. A. and Johnson, C. R. Topics in Matrix Analysis. Cambridge, England: Cambridge University Press, p. 208, 1994.Moler, C. and van Loan, C. "Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later." SIAM Rev. 45, 3-49, 2003.

Referenced on Wolfram|Alpha

Matrix Exponential

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Matrix Exponential." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MatrixExponential.html

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