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 be an real or complex matrix. The exponential of , denoted by or , is the matrix given by the power series
where formula_2 is defined to be the identity matrix formula_3 with the same dimensions as formula_4.
The above series always converges, so the exponential of is well-defined. If is a 1×1 matrix the matrix exponential of is a 1×1 matrix whose single element is the ordinary exponential of the single element of .
Properties.
Elementary properties.
Let and be complex matrices and let and be arbitrary complex numbers. We denote the identity matrix by 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_8 commute", it makes no difference to the argument whether formula_4 and formula_8 are numbers or matrices. It is important to note that this identity typically does not hold if formula_4 and formula_8 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 is symmetric then is also symmetric, and if is skew-symmetric then is orthogonal. If is Hermitian then is also Hermitian, and if is skew-Hermitian then is unitary.
Finally, 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 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 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:
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 , which implies that must be invertible.
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) and we know that the exponential function satisfies . The same is true for commuting matrices. If matrices and commute (meaning that ), 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
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:
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 and 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, is not guaranteed to be real for Hermitian , , . However, Lieb proved 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 is given by . 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 , 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 and ,
where ‖ · ‖ denotes an arbitrary matrix norm. It follows that the exponential map is continuous and Lipschitz continuous on compact subsets of .
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, for a generic -dependent exponent, ,
Taking the above expression 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,
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. 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:
then its exponential can be obtained by exponentiating each entry on the main diagonal:
This result also allows one to exponentiate diagonalizable matrices. If 
and is diagonal, then 
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 
can be diagonalized as 
Thus, 
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.
By the Jordan–Chevalley decomposition, any formula_36 matrix "X" with complex entries can be expressed as
where
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.
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
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 "J" is then given by
Projection case.
If is a projection matrix (i.e. is idempotent: ), its matrix exponential is:
Deriving this by expansion of the exponential function, each power of reduces to which becomes a common factor of the sum:
Rotation case.
For a simple rotation in which the perpendicular unit vectors and specify a plane, the rotation matrix can be expressed in terms of a similar exponential function involving a generator and angle .
The formula for the exponential results from reducing the powers of in the series expansion and identifying the respective series coefficients of and with and respectively. The second expression here for is the same as the expression for in the article containing the derivation of the generator, .
In two dimensions, if formula_46 and formula_47, then formula_48, formula_49, and
reduces to the standard matrix for a plane rotation.
The matrix projects a vector onto the -plane and the rotation only affects this part of the vector. An example illustrating this is a rotation of in the plane spanned by and ,
Let , so and its products with and are zero. This will allow us to evaluate powers of .
Evaluation by Laurent series.
By virtue of the Cayley–Hamilton theorem the matrix exponential is expressible as a polynomial of order −1.
If and are nonzero polynomials in one variable, such that , and if the meromorphic function
is entire, then
To prove this, multiply the first of the two above equalities by and replace by .
Such a polynomial can be found as follows−see Sylvester's formula. Letting be a root of , is solved from the product of by the principal part of the Laurent series of at : It is proportional to the relevant Frobenius covariant. Then the sum "St" of the "Qa,t", where runs over all the roots of , can be taken as a particular . All the other "Qt" will be obtained by adding a multiple of to . In particular, , the Lagrange-Sylvester polynomial, is the only whose degree is less than that of .
Example: Consider the case of an arbitrary 2×2 matrix,
The exponential matrix , by virtue of the Cayley–Hamilton theorem, must be of the form
Let and be the roots of the characteristic polynomial of ,
Then we have
hence
if ; while, if ,
so that
Defining
we have
where is 0 if = 0, and if = 0.
Thus,
Thus, as indicated above, the matrix 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 can also be given the following "interpolation" characterization. Define , and ≡ deg . Then is the unique degree polynomial which satisfies whenever is less than the multiplicity of as a root of . We assume, as we obviously can, that is the minimal polynomial of . We further assume that is a diagonalizable matrix. In particular, the roots of are simple, and the "interpolation" characterization indicates that is given by the Lagrange interpolation formula, so it is the Lagrange−Sylvester polynomial .
At the other extreme, if , then
The simplest case not covered by the above observations is when formula_67 with , 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 amounts to a linear combination of the first −1 powers of by the Cayley–Hamilton theorem. For diagonalizable matrices, as illustrated above, e.g. in the 2×2 case, Sylvester's formula yields , where the s are the Frobenius covariants of .
It is easiest, however, to simply solve for these s directly, by evaluating this expression and its first derivative at = 0, in terms of and , to find the same answer as above.
But this simple procedure also works for defective matrices, in a generalization due to Buchheim. This is illustrated here for a 4×4 example of a matrix which is "not diagonalizable", and the s are not projection matrices.
Consider
with eigenvalues and , each with a multiplicity of two.
Consider the exponential of each eigenvalue multiplied by , . Multiply each exponentiated eigenvalue by the corresponding undetermined coefficient matrix . If the eigenvalues have an algebraic multiplicity greater than 1, then repeat the process, but now multiplying by an extra factor of for each repetition, to ensure linear independence.
Sum all such terms, here four such,
To solve for all of the unknown matrices in terms of the first three powers of and the identity, one needs four equations, the above one providing one such at = 0. Further, differentiate it with respect to ,
and again,
and once more,
Setting = 0 in these four equations, the four coefficient matrices s may now be solved for,
to yield
Substituting with the value for 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"(λ"I" + "N") = "e"λ "e"N mentioned above; this yields
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 .
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 and multiplying throughout, yields
The second step is possible due to the fact that, if , then . So, calculating leads to the solution to the system, by simply integrating the third step with respect to .
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 ,
For to be a solution,
Thus,
where is determined by the initial conditions of the problem.
More precisely, consider the equation
with the initial condition , where
Left-multiplying the above displayed equality by yields
We claim that the solution to the equation
with the initial conditions formula_106 for is
where the notation is as follows:
 is the coefficient of formula_111 in the polynomial denoted by formula_112 in Subsection Evaluation by Laurent series above.
To justify this claim, we transform our order scalar equation into an order one vector equation by the usual reduction to a first order system. Our vector equation takes the form
where is the transpose companion matrix of . 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 = 2 we get the following statement. The solution to
is
where the functions and are as in Subsection Evaluation by Laurent series above.
Matrix-matrix exponentials.
The matrix exponential of another matrix (matrix-matrix exponential), is defined as
for any normal and non-singular matrix , and any complex matrix .
For matrix-matrix exponentials, there is a distinction between the left exponential and the right exponential , because the multiplication operator for matrix-to-matrix is not commutative. Moreover,