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Matrix Exponential

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The matrix exponential of a square matrix A is defined by the power series

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

which converges for every 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).

For a constant matrix H, the system ic^'(t)=Hc(t) has solution

c(t)=e^(-itH)c(0).
(5)

If H is a Hermitian matrix and t is real, the propagator is a unitary matrix and preserves the usual inner product. This construction is used for a finite vector basis approximation to the quartic oscillator (Trott 2006). Polynomial approximations to this exponential have been developed using Hermite polynomials and Gegenbauer polynomials (Vijay et al. 1999, Vijay and Metiu 2002). Parity-time symmetry alone does not give this unitarity property in the usual inner product.

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],
(6)

the matrix exponential is given by

exp(A)=[e^(a_1) 0 ... 0; 0 e^(a_2) ... 0; | | ... |; 0 0 ... e^(a_k)].
(7)

When a matrix is diagonalizable, its exponential can be computed by first diagonalizing 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],
(8)

then

exp(A)=[1 x z+1/2xy; 0 1 y; 0 0 1]
(9)

and A^3=0.

For the zero matrix A=0,

e^(0)=I,
(10)

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

e^(A)e^(-A)=e^(0)=I,
(11)

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)
(12)

holds whenever A and B commute, i.e.,

[A,B]=AB-BA=0.
(13)

The identity need not hold when A and B do not commute. For example,

exp([0 -x; 0 0]+[0 0; x 0])=[cosx -sinx; sinx cosx],
(14)

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].
(15)

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)],
(16)

where

m_(11)=e^((a+d)/2)[Deltacosh(1/2Delta)+(a-d)sinh(1/2Delta)]
(17)
m_(12)=2be^((a+d)/2)sinh(1/2Delta)
(18)
m_(21)=2ce^((a+d)/2)sinh(1/2Delta)
(19)
m_(22)=e^((a+d)/2)[Deltacosh(1/2Delta)+(d-a)sinh(1/2Delta)],
(20)

and Delta=sqrt((a-d)^2+4bc). 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 Differential Equation, 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.Numberphile. "A Number to the Power of a Matrix." Dec. 12, 2024. https://www.youtube.com/watch?v=CHozRTwHInE.Trott, M. "Ground-State of the Quartic Oscillator." §2.10 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.Vijay, A. and Metiu, H. "A Polynomial Expansion of the Quantum Propagator, the Green's Function, and the Spectral Density Operator." J. Chem. Phys. 116, 60-68, 2002. https://doi.org/10.1063/1.1425824.Vijay, A.; Wyatt, R. E.; and Billing, G. D. "Time Propagation and Spectral Filters in Quantum Dynamics: A Hermite Polynomial Perspective." J. Chem. Phys. 111, 10794-10805, 1999. https://doi.org/10.1063/1.480483.

Referenced on Wolfram|Alpha

Matrix Exponential

Cite this as:

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

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