# Matrix exponential

In mathematics, the **matrix exponential** is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group.

Let X be an *n*×*n* real or complex matrix. The exponential of X, denoted by *e*^{X} or exp(*X*), is the *n*×*n* matrix given by the power series

where is defined to be the identity matrix with the same dimensions as .[1]

The above series always converges, so the exponential of X is well-defined. If X is a 1×1 matrix the matrix exponential of X is a 1×1 matrix whose single element is the ordinary exponential of the single element of X.

## Properties

### Elementary properties

Let *X* and *Y* be *n*×*n* complex matrices and let *a* and *b* be arbitrary complex numbers. We denote the *n*×*n* identity matrix by *I* and the zero matrix by 0. The matrix exponential satisfies the following properties.[2]

We begin with the properties that are immediate consequences of the definition as a power series:

*e*^{0}=*I*- exp(
*X*^{T}) = (exp*X*)^{T}, where*X*^{T}denotes the transpose of*X*. - exp(
*X*^{∗}) = (exp*X*)^{∗}, where*X*^{∗}denotes the conjugate transpose of*X*. - If
*Y*is invertible then*e*^{YXY−1}=*Ye*^{X}*Y*^{−1}.

The next key result is this one:

- If then .

The proof of this identity is the same as the standard power-series argument for the corresponding identity for the exponential of real numbers. That is to say, *as long as and commute*, it makes no difference to the argument whether and are numbers or matrices. It is important to note that this identity typically does not hold if and do not commute (see Golden-Thompson inequality below).

Consequences of the preceding identity are the following:

*e*^{aX}*e*^{bX}=*e*^{(a + b)X}*e*^{X}*e*^{−X}=*I*

Using the above results, we can easily verify the following claims. If *X* is symmetric then *e*^{X} is also symmetric, and if *X* is skew-symmetric then *e*^{X} is orthogonal. If *X* is Hermitian then *e*^{X} is also Hermitian, and if *X* is skew-Hermitian then *e*^{X} is unitary.

Finally, we have the following:

- A Laplace transform of matrix exponentials amounts to the resolvent,

- for all sufficiently large positive values of
*s*.

### Linear differential equation systems

One of the reasons for the importance of the matrix exponential is that it can be used to solve systems of linear ordinary differential equations. The solution of

where A is a constant matrix, is given by

The matrix exponential can also be used to solve the inhomogeneous equation

See the section on applications below for examples.

There is no closed-form solution for differential equations of the form

where A is not constant, but the Magnus series gives the solution as an infinite sum.

### The determinant of the matrix exponential

By Jacobi's formula, for any complex square matrix the following trace identity holds:[3]

In addition to providing a computational tool, this formula demonstrates that a matrix exponential is always an invertible matrix. This follows from the fact that the right hand side of the above equation is always non-zero, and so det(*e ^{A}*) ≠ 0, which implies that

*e*must be invertible.

^{A}In the real-valued case, the formula also exhibits the map

to not be surjective, in contrast to the complex case mentioned earlier. This follows from the fact that, for real-valued matrices, the right-hand side of the formula is always positive, while there exist invertible matrices with a negative determinant.

## The exponential of sums

For any real numbers (scalars) x and y we know that the exponential function satisfies *e*^{x+y} = *e*^{x} *e*^{y}. The same is true for commuting matrices. If matrices X and Y commute (meaning that *XY* = *YX*), then,

However, for matrices that do not commute the above equality does not necessarily hold.

### The Lie product formula

Even if and do not commute, the exponential can be computed by the Lie product formula[4]

- .

### The Baker–Campbell–Hausdorff formula

In the other direction, if and are sufficiently small (but not necessarily commuting) matrices, we have

where may be computed as a series in commutators of and by means of the Baker–Campbell–Hausdorff formula:[5]

- ,

where the remaining terms are all iterated commutators involving and . If and commute, then all the commutators are zero and we have simply .

## Inequalities for exponentials of Hermitian matrices

For Hermitian matrices there is a notable theorem related to the trace of matrix exponentials.

If A and B are Hermitian matrices, then

There is no requirement of commutativity. There are counterexamples to show that the Golden–Thompson inequality cannot be extended to three matrices – and, in any event, tr(exp(*A*)exp(*B*)exp(*C*)) is not guaranteed to be real for Hermitian *A*, *B*, *C*. However, Lieb proved[7][8]
that it can be generalized to three matrices if we modify the expression as follows

## The exponential map

The exponential of a matrix is always an invertible matrix. The inverse matrix of *e*^{X} is given by *e*^{−X}. This is analogous to the fact that the exponential of a complex number is always nonzero. The matrix exponential then gives us a map

from the space of all *n*×*n* matrices to the general linear group of degree n, i.e. the group of all *n*×*n* invertible matrices. In fact, this map is surjective which means that every invertible matrix can be written as the exponential of some other matrix[9] (for this, it is essential to consider the field **C** of complex numbers and not **R**).

For any two matrices X and Y,

where || · || denotes an arbitrary matrix norm. It follows that the exponential map is continuous and Lipschitz continuous on compact subsets of *M*_{n}(**C**).

The map

defines a smooth curve in the general linear group which passes through the identity element at *t* = 0.

In fact, this gives a one-parameter subgroup of the general linear group since

The derivative of this curve (or tangent vector) at a point *t* is given by

The derivative at *t* = 0 is just the matrix *X*, which is to say that *X* generates this one-parameter subgroup.

More generally,[10] for a generic t-dependent exponent, *X(t)*,

Taking the above expression *e*^{X(t)} outside the integral sign and expanding the integrand with the help of the Hadamard lemma one can obtain the following useful expression for the derivative of the matrix exponent,[11]

The coefficients in the expression above are different from what appears in the exponential. For a closed form, see derivative of the exponential map.

## Computing the matrix exponential

Finding reliable and accurate methods to compute the matrix exponential is difficult, and this is still a topic of considerable current research in mathematics and numerical analysis. Matlab, GNU Octave, and SciPy all use the Padé approximant.[12][13][14] In this section, we discuss methods that are applicable in principle to any matrix, and which can be carried out explicitly for small matrices.[15] Subsequent sections describe methods suitable for numerical evaluation on large matrices.

### Diagonalizable case

If a matrix is diagonal:

- ,

then its exponential can be obtained by exponentiating each entry on the main diagonal:

- .

This result also allows one to exponentiate diagonalizable matrices. If

*A*=*UDU*^{−1}

and *D* is diagonal, then

*e*^{A}=*Ue*^{D}*U*^{−1}.

Application of Sylvester's formula yields the same result. (To see this, note that addition and multiplication, hence also exponentiation, of diagonal matrices is equivalent to element-wise addition and multiplication, and hence exponentiation; in particular, the "one-dimensional" exponentiation is felt element-wise for the diagonal case.)

### Nilpotent case

A matrix *N* is nilpotent if *N*^{q} = 0 for some integer *q*. In this case, the matrix exponential *e*^{N} can be computed directly from the series expansion, as the series terminates after a finite number of terms:

### General case

#### Using the Jordan–Chevalley decomposition

Any matrix *X* with complex entries can be expressed as

where

*A*is diagonalizable*N*is nilpotent*A*commutes with*N*(i.e.*AN*=*NA*)

This is the Jordan–Chevalley decomposition.

This means that we can compute the exponential of *X* by reducing to the previous two cases:

Note that we need the commutativity of *A* and *N* for the last step to work.

#### Using the Jordan canonical form

Another (closely related) method if the field is algebraically closed is to work with the Jordan form of *X*. Suppose that *X* = *PJP*^{ −1} where *J* is the Jordan form of *X*. Then

Also, since

Therefore, we need only know how to compute the matrix exponential of a Jordan block. But each Jordan block is of the form

where *N* is a special nilpotent matrix. The matrix exponential of this block is given by

### Projection case

If *P* is a projection matrix (i.e. is idempotent: P^{2}=P, so multiplying P by itself any number of times is itself), its matrix exponential is *e*^{P} = *I* + (*e* − 1)*P*. This may be derived by expansion of the definition of the exponential function and by use of the idempotency of *P*:

### Rotation case

For a simple rotation in which the perpendicular unit vectors a and b specify a plane,[16] the rotation matrix R can be expressed in terms of a similar exponential function involving a generator G and angle θ.[17][18]

The formula for the exponential results from reducing the powers of G in the series expansion and identifying the respective series coefficients of *G ^{2}* and G with −cos(

*θ*) and sin(

*θ*) respectively. The second expression here for

*e*is the same as the expression for

^{Gθ}*R*(

*θ*) in the article containing the derivation of the generator,

*R*(

*θ*) =

*e*.

^{Gθ}In two dimensions, if and , then , , and

reduces to the standard matrix for a plane rotation.

The matrix *P* = −*G*^{2} projects a vector onto the ab-plane and the rotation only affects this part of the vector. An example illustrating this is a rotation of 30° = π/6 in the plane spanned by *a* and *b*,

Let *N* = *I* − *P*, so *N*^{2} = *N* and its products with *P* and *G* are zero. This will allow us to evaluate powers of *R*.

## Evaluation by Laurent series

By virtue of the Cayley–Hamilton theorem the matrix exponential is expressible as a polynomial of order n−1.

If P and *Q _{t}* are nonzero polynomials in one variable, such that

*P*(

*A*) = 0, and if the meromorphic function

is entire, then

- .

To prove this, multiply the first of the two above equalities by *P*(*z*) and replace z by A.

Such a polynomial *Q _{t}(z)* can be found as follows−−see Sylvester's formula. Letting a be a root of P,

*Q*is solved from the product of P by the principal part of the Laurent series of f at a: It is proportional to the relevant Frobenius covariant. Then the sum

_{a,t}(z)*S*of the

_{t}*Q*, where a runs over all the roots of P, can be taken as a particular

_{a,t}*Q*. All the other

_{t}*Q*will be obtained by adding a multiple of P to

_{t}*S*. In particular,

_{t}(z)*S*, the Lagrange-Sylvester polynomial, is the only

_{t}(z)*Q*whose degree is less than that of P.

_{t}**Example**: Consider the case of an arbitrary 2-by-2 matrix,

The exponential matrix e^{tA}, by virtue of the Cayley–Hamilton theorem, must be of the form

- .

(For any complex number z and any * C*-algebra B, we denote again by z the product of z by the unit of B.)

Let α and β be the roots of the characteristic polynomial of A,

Then we have

hence

if *α* ≠ *β*; while, if *α* = *β*,

so that

Defining

we have

where sin(*qt*)/*q* is 0 if t = 0, and t if q = 0.
Thus,

Thus, as indicated above, the matrix A having decomposed into the sum of two mutually commuting pieces, the traceful piece and the traceless piece,

the matrix exponential reduces to a plain product of the exponentials of the two respective pieces. This is a formula often used in physics, as it amounts to the analog of Euler's formula for Pauli spin matrices, that is rotations of the doublet representation of the group SU(2).

The polynomial *S _{t}* can also be given the following "interpolation" characterization. Define

*e*, and n ≡ degP. Then

_{t}(z) ≡ e^{tz}*S*is the unique degree <

_{t}(z)*n*polynomial which satisfies

*S*=

_{t}^{(k)}(a)*e*whenever k is less than the multiplicity of a as a root of P. We assume, as we obviously can, that P is the minimal polynomial of A. We further assume that A is a diagonalizable matrix. In particular, the roots of P are simple, and the "interpolation" characterization indicates that

_{t}^{(k)}(a)*S*is given by the Lagrange interpolation formula, so it is the Lagrange−Sylvester polynomial .

_{t}At the other extreme, if *P* = *(z−a) ^{n}*, then

The simplest case not covered by the above observations is when with *a* ≠ *b*, which yields

## Evaluation by implementation of Sylvester's formula

A practical, expedited computation of the above reduces to the following rapid steps.
Recall from above that an *n×n* matrix exp(*tA*) amounts to a linear combination of the first n−1 powers of A by the Cayley–Hamilton theorem. For diagonalizable matrices, as illustrated above, e.g. in the 2×2 case, Sylvester's formula yields exp(*tA*) = *B _{α}* exp(

*tα*)+

*B*exp(

_{β}*tβ*), where the Bs are the Frobenius covariants of A.

It is easiest, however, to simply solve for these Bs directly, by evaluating this expression and its first derivative at t=0, in terms of A and I, to find the same answer as above.

But this simple procedure also works for defective matrices, in a generalization due to Buchheim.[19] This is illustrated here for a 4×4 example of a matrix which is *not diagonalizable*, and the Bs are not projection matrices.

Consider

with eigenvalues *λ*_{1}=3/4 and *λ*_{2}=1, each with a
multiplicity of two.

Consider the exponential of each eigenvalue multiplied by t, exp(*λ _{i}t*). Multiply each exponentiated eigenvalue by the corresponding undetermined coefficient matrix

*B*

_{i}. If the eigenvalues have an algebraic multiplicity greater than 1, then repeat the process, but now multiplying by an extra factor of t for each repetition, to ensure linear independence.

(If one eigenvalue had a multiplicity of three, then there would be the three terms: . By contrast, when all eigenvalues are distinct, the Bs are just the Frobenius covariants, and solving for them as below just amounts to the inversion of the Vandermonde matrix of these 4 eigenvalues.)

Sum all such terms, here four such,

- .

To solve for all of the unknown matrices B in terms of the first three powers of A and the identity, one needs four equations, the above one providing one such at t =0. Further, differentiate it with respect to t,

and again,

and once more,

- .

(In the general case, n−1 derivatives need be taken.)

Setting t=0 in these four equations, the four coefficient matrices Bs may now be solved for,

- ,

to yield

- .

Substituting with the value for A yields the coefficient matrices

so the final answer is

- .

The procedure is much shorter than Putzer's algorithm sometimes utilized in such cases.

## Illustrations

Suppose that we want to compute the exponential of

Its Jordan form is

where the matrix *P* is given by

Let us first calculate exp(*J*). We have

The exponential of a 1×1 matrix is just the exponential of the one entry of the matrix, so exp(*J*_{1}(4)) = [*e*^{4}]. The exponential of *J*_{2}(16) can be calculated by the formula *e*^{(λI + N)} = *e*^{λ} *e*^{N} mentioned above; this yields[20]

Therefore, the exponential of the original matrix *B* is

## Applications

### Linear differential equations

The matrix exponential has applications to systems of linear differential equations. (See also matrix differential equation.) Recall from earlier in this article that a *homogeneous* differential equation of the form

has solution *e*^{At} * y*(0).

If we consider the vector

we can express a system of *inhomogeneous* coupled linear differential equations as

Making an ansatz to use an integrating factor of *e*^{−At} and multiplying throughout, yields

The second step is possible due to the fact that, if *AB* = *BA*, then *e*^{At}*B* = *Be*^{At}. So, calculating *e*^{At} leads to the solution to the system, by simply integrating the third step with respect to t.

#### Example (homogeneous)

Consider the system

The associated defective matrix is

The matrix exponential is

so that the general solution of the homogeneous system is

amounting to

#### Example (inhomogeneous)

Consider now the inhomogeneous system

We again have

and

From before, we already have the general solution to the homogeneous equation. Since the sum of the homogeneous and particular solutions give the general solution to the inhomogeneous problem, we now only need find the particular solution.

We have, by above,

which could be further simplified to get the requisite particular solution determined through variation of parameters.
Note **c** = **y**_{p}(0). For more rigor, see the following generalization.

### Inhomogeneous case generalization: variation of parameters

For the inhomogeneous case, we can use integrating factors (a method akin to variation of parameters). We seek a particular solution of the form **y**_{p}(*t*) = exp(*tA*) **z** (*t*) ,

For **y**_{p} to be a solution,

Thus,

where * c* is determined by the initial conditions of the problem.

More precisely, consider the equation

with the initial condition *Y(t _{0})* =

*Y*, where A is an n by n complex matrix,

_{0}F is a continuous function from some open interval I to ℂ^{n},

is a point of I, and

is a vector of ℂ^{n}.

Left-multiplying the above displayed equality by *e ^{−tA}* yields

We claim that the solution to the equation

with the initial conditions for 0 ≤ *k < n* is

where the notation is as follows:

is a monic polynomial of degree *n* > 0,

f is a continuous complex valued function defined on some open interval I,

is a point of I,

is a complex number, and

*s _{k}(t)* is the coefficient of in the polynomial denoted by in Subsection Evaluation by Laurent series above.

To justify this claim, we transform our order n scalar equation into an order one vector equation by the usual reduction to a first order system. Our vector equation takes the form

where A is the transpose companion matrix of P. We solve this equation as explained above, computing the matrix exponentials by the observation made in Subsection Evaluation by implementation of Sylvester's formula above.

In the case n = 2 we get the following statement. The solution to

is

where the functions *s*_{0} and *s*_{1} are as in Subsection Evaluation by Laurent series above.

## Matrix-matrix exponentials

The matrix exponential of another matrix (matrix-matrix exponential),[21] is defined as

for X any normal and non-singular *n*×*n* matrix, and Y any complex *n*×*n* matrix.

For matrix-matrix exponentials, there is a distinction between the left exponential ^{Y}X and the right exponential X^{Y}, because the multiplication operator for matrix-to-matrix is not commutative. Moreover,

- If X is normal and non-singular, then X
^{Y}and^{Y}X have the same set of eigenvalues. - If X is normal and non-singular, Y is normal, and
*XY*=*YX*, then*X*=^{Y}.^{Y}X - If X is normal and non-singular, and X, Y, Z commute with each other, then
*X*=^{Y+Z}*X*·^{Y}*X*and^{Z}=^{Y+Z}X·^{Y}X.^{Z}X

## See also

- Matrix function
- Matrix logarithm
- Exponential function
- Exponential map (Lie theory)
- Magnus expansion
- Derivative of the exponential map
- Vector flow
- Golden–Thompson inequality
- Phase-type distribution
- Lie product formula
- Baker–Campbell–Hausdorff formula
- Frobenius covariant
- Sylvester's formula
- Trigonometric functions of matrices

## References

- Hall 2015 Equation 2.1
- Hall 2015 Proposition 2.3
- Hall 2015 Theorem 2.12
- Hall 2015 Theorem 2.11
- Hall 2015 Chapter 5
- Bhatia, R. (1997).
*Matrix Analysis*. Graduate Texts in Mathematics.**169**. Springer. ISBN 978-0-387-94846-1. - E. H. Lieb (1973). "Convex trace functions and the Wigner–Yanase–Dyson conjecture".
*Advances in Mathematics*.**11**(3): 267–288. doi:10.1016/0001-8708(73)90011-X. - H. Epstein (1973). "Remarks on two theorems of E. Lieb".
*Communications in Mathematical Physics*.**31**(4): 317–325. Bibcode:1973CMaPh..31..317E. doi:10.1007/BF01646492. - Hall 2015 Exercises 2.9 and 2.10
- R. M. Wilcox (1967). "Exponential Operators and Parameter Differentiation in Quantum Physics".
*Journal of Mathematical Physics*.**8**(4): 962–982. Bibcode:1967JMP.....8..962W. doi:10.1063/1.1705306. - Hall 2015 Theorem 5.4
- "Matrix exponential – MATLAB expm – MathWorks Deutschland". Mathworks.de. 2011-04-30. Retrieved 2013-06-05.
- "GNU Octave – Functions of a Matrix". Network-theory.co.uk. 2007-01-11. Retrieved 2013-06-05.
- "scipy.linalg.expm function documentation". The SciPy Community. 2015-01-18. Retrieved 2015-05-29.
- See Hall 2015 Section 2.2
- in a Euclidean space
- Weyl, Hermann (1952).
*Space Time Matter*. Dover. p. 142. ISBN 978-0-486-60267-7. - Bjorken, James D.; Drell, Sidney D. (1964).
*Relativistic Quantum Mechanics*. McGraw-Hill. p. 22. - Rinehart, R. F. (1955). "The equivalence of definitions of a matric function".
*The American Mathematical Monthly*,**62**(6), 395-414. - This can be generalized; in general, the exponential of
*J*_{n}(*a*) is an upper triangular matrix with*e*^{a}/0! on the main diagonal,*e*^{a}/1! on the one above,*e*^{a}/2! on the next one, and so on. - Ignacio Barradas and Joel E. Cohen (1994). "Iterated Exponentiation, Matrix-Matrix Exponentiation, and Entropy" (PDF). Academic Press, Inc. Archived from the original (PDF) on 2009-06-26.

- Hall, Brian C. (2015),
*Lie groups, Lie algebras, and representations: An elementary introduction*, Graduate Texts in Mathematics,**222**(2nd ed.), Springer, ISBN 978-3-319-13466-6 - Horn, Roger A.; Johnson, Charles R. (1991).
*Topics in Matrix Analysis*. Cambridge University Press. ISBN 978-0-521-46713-1.. - Moler, Cleve; Van Loan, Charles F. (2003). "Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later" (PDF).
*SIAM Review*.**45**(1): 3–49. Bibcode:2003SIAMR..45....3M. CiteSeerX 10.1.1.129.9283. doi:10.1137/S00361445024180. ISSN 1095-7200.. - Suzuki, Masuo (1985). "Decomposition formulas of exponential operators and Lie exponentials with some applications to quantum mechanics and statistical physics".
*Journal of Mathematical Physics*.**26**(4): 601–612. Bibcode:1985JMP....26..601S. doi:10.1063/1.526596. - Curtright, T L; Fairlie, D B; Zachos, C K (2014). "A compact formula for rotations as spin matrix polynomials".
*Symmetry, Integrability and Geometry: Methods and Applications*.**10**: 084. arXiv:1402.3541. Bibcode:2014SIGMA..10..084C. doi:10.3842/SIGMA.2014.084. - Householder, Alston S. (2006).
*The Theory of Matrices in Numerical Analysis*. Dover Books on Mathematics. ISBN 978-0-486-44972-2. - Van Kortryk, T. S. (2016). "Matrix exponentials, SU(N) group elements, and real polynomial roots".
*Journal of Mathematical Physics*.**57**(2): 021701. arXiv:1508.05859. Bibcode:2016JMP....57b1701V. doi:10.1063/1.4938418.