Matrix operation generalizing exponentiation of scalar numbers
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 exponential map 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
formula_1
where formula_2 is defined to be the identity matrix formula_3 with the same dimensions as formula_4. The series always converges, so the exponential of X is well-defined.
Equivalently,
formula_5
where I is the "n"×"n" identity matrix. 
When X is an "n"×"n" diagonal matrix then exp("X") will be an "n"×"n" diagonal matrix with each diagonal element equal to the ordinary exponential applied to the corresponding diagonal 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.
We begin with the properties that are immediate consequences of the definition as a power series:
The next key result is this one:
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 formula_4 and formula_9 commute", it makes no difference to the argument whether formula_4 and formula_9 are numbers or matrices. It is important to note that this identity typically does not hold if formula_4 and formula_9 do not commute (see Golden-Thompson inequality below).
Consequences of the preceding identity are the following:
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, a Laplace transform of matrix exponentials amounts to the resolvent,
formula_14
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
formula_15
where A is a constant matrix and "y" is a column vector, is given by
formula_16
The matrix exponential can also be used to solve the inhomogeneous equation
formula_17
See the section on applications below for examples.
There is no closed-form solution for differential equations of the form
formula_18
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:
formula_19
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("eA") ≠ 0, which implies that "eA" must be invertible.
In the real-valued case, the formula also exhibits the map
formula_20
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.
Real symmetric matrices.
The matrix exponential of a real symmetric matrix is positive definite. Let formula_21 be an "n"×"n" real symmetric matrix and formula_22 a column vector. Using the elementary properties of the matrix exponential and of symmetric matrices, we have:
formula_23
Since formula_24 is invertible, the equality only holds for formula_25, and we have formula_26 for all non-zero formula_27. Hence formula_28 is positive definite.
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,
formula_29
However, for matrices that do not commute the above equality does not necessarily hold.
The Lie product formula.
Even if X and Y do not commute, the exponential "e""X" + "Y" can be computed by the Lie product formula
formula_30
Using a large finite k to approximate the above is basis of the Suzuki-Trotter expansion, often used in numerical time evolution.
The Baker–Campbell–Hausdorff formula.
In the other direction, if X and Y are sufficiently small (but not necessarily commuting) matrices, we have
formula_31
where Z may be computed as a series in commutators of X and Y by means of the Baker–Campbell–Hausdorff formula:
formula_32
where the remaining terms are all iterated commutators involving X and Y. If X and Y commute, then all the commutators are zero and we have simply "Z" = "X" + "Y".
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
formula_33
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 that it can be generalized to three matrices if we modify the expression as follows
formula_34
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
formula_35
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 (for this, it is essential to consider the field C of complex numbers and not R).
For any two matrices X and Y,
formula_36
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
formula_37
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
formula_38
The derivative of this curve (or tangent vector) at a point "t" is given by
 ;templatestyles src="Numbered block/styles.css" / ;
The derivative at "t" = 0 is just the matrix "X", which is to say that "X" generates this one-parameter subgroup.
More generally, for a generic t-dependent exponent, "X"("t"),
formula_39
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,
formula_40
The coefficients in the expression above are different from what appears in the exponential. For a closed form, see derivative of the exponential map.
Directional derivatives when restricted to Hermitian matrices.
Let formula_4 be a formula_42 Hermitian matrix with distinct eigenvalues. Let formula_43 be its eigen-decomposition where formula_44 is a unitary matrix whose columns are the eigenvectors of formula_4, formula_46 is its conjugate transpose, and formula_47 the vector of corresponding eigenvalues. Then, for any formula_42 Hermitian matrix formula_49, the directional derivative of formula_50 at formula_4 in the direction formula_49 is
formula_53
where formula_54, the operator formula_55 denotes the Hadamard product, and, for all formula_56, the matrix formula_57 is defined as
formula_58
In addition, for any formula_42 Hermitian matrix formula_60, the second directional derivative in directions formula_60 and formula_49 is
formula_63
where the matrix-valued function formula_64 is defined, for all formula_56, as
formula_66
with
formula_67
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, R, and SciPy all use the Padé approximant. In this section, we discuss methods that are applicable in principle to any matrix, and which can be carried out explicitly for small matrices. Subsequent sections describe methods suitable for numerical evaluation on large matrices.
Diagonalizable case.
If a matrix is diagonal:
formula_68
then its exponential can be obtained by exponentiating each entry on the main diagonal:
formula_69
This result also allows one to exponentiate diagonalizable matrices. If
 ;templatestyles src="Block indent/styles.css"/ ;"A" = "UDU"−1
and "D" is diagonal, then
 ;templatestyles src="Block indent/styles.css"/ ;"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.)
Example : Diagonalizable.
For example, the matrix
formula_70
can be diagonalized as
formula_71
Thus,
formula_72
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:
formula_73
Since the series has a finite number of steps, it is a matrix polynomial, which can be computed efficiently.
General case.
Using the Jordan–Chevalley decomposition.
By the Jordan–Chevalley decomposition, any formula_42 matrix "X" with complex entries can be expressed as
formula_75
where
This means that we can compute the exponential of "X" by reducing to the previous two cases:
formula_76
Note that we need the commutativity of "A" and "N" for the last step to work.
Using the Jordan canonical form.
A closely related method is, if the field is algebraically closed, to work with the Jordan form of X. Suppose that "X" = "PJP"−1 where J is the Jordan form of X. Then
formula_77
Also, since
formula_78
Therefore, we need only know how to compute the matrix exponential of a Jordan block. But each Jordan block is of the form
formula_79
where N is a special nilpotent matrix. The matrix exponential of J is then given by
formula_80
Projection case.
If P is a projection matrix (i.e. is idempotent: "P"2 = "P"), its matrix exponential is:
 ;templatestyles src="Block indent/styles.css"/ ;"e""P" = "I" + ("e" − 1)"P".
Deriving this by expansion of the exponential function, each power of P reduces to P which becomes a common factor of the sum:
formula_81
Rotation case.
For a simple rotation in which the perpendicular unit vectors a and b specify a plane, the rotation matrix R can be expressed in terms of a similar exponential function involving a generator G and angle θ.
formula_82
formula_83
The formula for the exponential results from reducing the powers of G in the series expansion and identifying the respective series coefficients of "G2" and G with −cos("θ") and sin("θ") respectively. The second expression here for "eGθ" is the same as the expression for "R"("θ") in the article containing the derivation of the generator, "R"("θ") = "eGθ".
In two dimensions, if formula_84 and formula_85, then formula_86, formula_87, and
formula_88
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,
formula_89
formula_90
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".
formula_91
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 "Qt" are nonzero polynomials in one variable, such that "P"("A") = 0, and if the meromorphic function
formula_92
is entire, then
formula_93
To prove this, multiply the first of the two above equalities by "P"("z") and replace z by A.
Such a polynomial "Qt"("z") can be found as follows−see Sylvester's formula. Letting a be a root of P, "Qa,t"("z") 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 "St" of the "Qa,t", where a runs over all the roots of P, can be taken as a particular "Qt". All the other "Qt" will be obtained by adding a multiple of P to "St"("z"). In particular, "St"("z"), the Lagrange-Sylvester polynomial, is the only "Qt" whose degree is less than that of P.
Example: Consider the case of an arbitrary 2×2 matrix,
formula_94
The exponential matrix e"tA", by virtue of the Cayley–Hamilton theorem, must be of the form
formula_95
Let α and β be the roots of the characteristic polynomial of A,
formula_96
Then we have
formula_97
hence
formula_98
if "α" ≠ "β"; while, if "α" = "β",
formula_99
so that
formula_100
Defining
formula_101
we have
formula_102
where sin("qt")/"q" is 0 if "t" = 0, and t if "q" = 0.
Thus,
formula_103
Thus, as indicated above, the matrix A having decomposed into the sum of two mutually commuting pieces, the traceful piece and the traceless piece,
formula_104
the matrix exponential reduces to a plain product