Matrix Exponential

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


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)

(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],

The matrix exponential is given by

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

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],


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

and A^3=0.

For the zero matrix A=0,


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


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


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


For example,

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


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

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)]





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)].

See also

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

Portions of this entry contributed by Todd Rowland

Explore with Wolfram|Alpha


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.

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