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On the Exponentials of Some Structured Matrices

In this note explicit algorithms for calculating the exponentials of important structured 4 x 4 matrices are provided. These lead to closed form formulae for these exponentials. The techniques rely on one particular Clifford Algebra isomorphism and basic Lie theory. When used in conjunction with structure preserving similarities, such as Givens rotations, these techniques extend to dimensions bigger than four.

Finding matrix exponentials is arguably one of the most important goals of mathematical physics. In full generality, this is a thankless task, [1]. However, for matrices with structure, finding exponentials ought to be more tractable. In this note, confirmation of this phenomenon is given for a large class of 4 × 4 matrices with structure. These include skew-Hamiltonian, perskewsymmetric, bisymmetric (i.e., simultaneously symmetric and persymmetric -e.g., symmetric, Toeplitz), symmetric and Hamiltonian etc., Some of the techniques presented extend almost verbatim to some families of complex matrices (see Remark (3.4), for instance). Since such matrices arise in a variety of physical applications in both classical and quantum physics, it is interesting that their exponentials can be calculated algorithmically (these lead to closed form formulae), for the most part, without any auxiliary information about their spectrum.
For general symmetric matrices, however, the spectral decomposition of a 3 × 3 matrix is needed [see, however, iii) of Remark (2.1)]. On the other hand, this spectral decomposition can itself be produced in closed form. Thus, even for such matrices the techniques described here can be justifiably called closed form methods. For brevity, this note only records explicit algorithms for finding these matrix exponentials -the resultant final formulae can easily be written down once the reported procedures are implemented.
The methods discussed below are of two types. The first, which is more versatile, relies on an algebra isomorphism of real 4 × 4 matrices with H ⊗ H. This algebra isomorphism, known from the theory of Clifford algebras and which ought to be widely advertised, was used in a series of interesting articles by Mackey et al., [2,3,4] for finding eigenvalues of some of the structured matrices discussed here. The present note can be seen as a contribution of a similar type. It is emphasized that for the preponderance of the matrices, considered here, this algebra isomorphism alone is needed. In particular, in this note no use is made of any of the structure preserving rotations used in [2,3,4] ever -see ii) of Remark (3.2). The second is based on the observation that several "covering" space Lie group homomorphisms, when made explicit, contain in them a recipe for finding exponentials of matrices belonging to certain Lie algebras. This circumstance renders the exponentiating of some 4 × 4 matrices (real/complex) equivalent to the job of finding the exponential of 2×2 matrices -which can be done in closed form. This method is, however, applicable only to a limited family of matrices.
Therefore, this method is presented in an appendix.
It is worth noting that, though most of the structured matrices considered here were chosen for their importance in applications, the real enabling structure is that present in H ⊗ H. This is especially illustrated by certain normal matrices [see Definition (3.1)].
The balance of this note is organized as follows. In the next section some notation and one observation which is used throughout is recorded. In the same section the relation between H ⊗ H and gl(4, R) is presented. The third section discusses a wide family of matrices which can be exponentiated using the aformentioned algebra isomorphism. The final section offers conclusions. In the first appendix the second approach to exponentiating matrices in p(4, R) and so(2, 2, R) (see next section for notation) is presented in a manner that makes the connection to the covering space homomorphism between SU (2) × SU (2) and SO(4, R) explicit (see Remark (5.1)). In the final section thirteen classes of matrices are listed which can be exponentiated by mimicking verbatim two situations studied earlier.
In closing this introductory section it is noted that by combining these techniques with techniques such as structure preserving similarites, e.g, Givens rotations, [5], one can extend these results, in many cases, to find algorithmically the exponentials of structured matrices of size bigger than four. In other words, one can use such similarities (normally used in the literature for reduction to canonical forms) to reduce the exponential calculation to dimension four or lower of matrices with similar structure. In principle, this would provide closed form formulae for the exponentials of such structured matrices, since one can explicitly write down the desired Givens type similarities. However, it is more accurate to say that this implies an algorithmic procedure for exponentiating such matrices. For matrices for which this is possible (e.g., symmetric matrices), the details of this procedure is routine and hence will not be pursued here.

Notation and Preliminary Observations
The following definitions and notations will be frequently met in this work: • gl(n, R) and gl(n, C) represent the real (resp. complex) n × n matrices.
• sl(n, R) and sl(n, C) represent the real (resp. complex) traceless matrices. SL(n, R) and SL(n, C) represent the real (resp. complex) matrices of determinant one.
• SU (n) represents the n × n unitary matrices of determinant one. su(n) represents the n × n skew-Hermitian, traceless matrices.
• R n represents the matrix with 1 on the anti-diagonal and zeroes elsewhere. p(n, R) and p(n, C) represent the n×n real (resp. complex) matrices, A, satisfying A T R n +R n A = 0.
These matrices are also said to be perskewsymmetric. Persymmetric matrices are those matrices, X, which satisfy X T R n = R n X. P (n, R) (respectively P (n, C)) is the set of matrices (real/complex), X, which satisfy X T R n X = R n .
• J 2n is the 2n × 2n matrix which, in block form, is given by sp(2n, R) and sp(2n, C) represent those real (resp. complex) 2n × 2n matrices which satisfy X T J 2n + J 2n X = 0. Such matrices are also called Hamiltonian. Matrices, Z, satsifying Z T J 2n = J 2n Z are called skew-Hamiltonian.
so(p, q, R) and so(p, q, C) represent the real (resp. complex) n×n matrices (n = p + q), X, satisfying X T I p,q + I p,q X = 0.
• The anti-trace of an n × n matrix is the sum of the elements on its anti-diagonal. X, n × n, is anti-scalar if X = γR n , with γ ∈ R (or C).
• Throughout H will be denote the quaternions, while P stands for the purely imaginary quaternions, tacitly identified with R 3 .

Remark 2.1
• i) Throughout this note, use of the following observation will be made: Let X be an n× n matrix satsifying X 2 + c 2 I n = 0, c = 0. Then e X = cos(c)I n + sin(c) c X.
Here c 2 is allowed to be complex, and c is then taken to be √ re i θ 2 , with c 2 = re iθ , θ ∈ [0, 2π).
• Occassionally the fact that any matrix which satisfies X 3 = −c 2 X, c = 0, satisfies c X 2 (the Rodrigues's formula) will also be used.
• iii) Explicit formulae for e A can be produced if the minimal polynomial of A is known and it is low in degree (cf., [6] where such formulae are written down from the characteristic polynomial). Since it is possible to find the minimal polynomial of many of the matrices considered here explicitly (i.e., without any spectral information), this removes the need for the spectral decomposition, mentioned in the introduction, for A symmetric. However, since the corresponding explicit formulae for e A are more complicated than the ones in i) and ii), they will not be pursued here. See the conclusions for an illustration of this issue.
H ⊗ H and gl(4, R): • Associate to each product tensor p ⊗ q ∈ H ⊗ H, the matrix, M p,q , of the map which sends x ∈ H to pxq, identifying R 4 with H via this basis {1, i, j, k}, Extending this to the full tensor product by linearity, yields an algebra isomorphism This connection, which is known from the theory of Clifford Algebras, has been put to great practical use in solving eigenvalue problems for structured matrices by Mackey et al., [2,3,4]. It can also be used for finding exponentials, e A , A ∈ gl(4, R) via the following procedure: General Algorithm for e A Using H ⊗ H • I) Identify u ∈ H ⊗ H, corresponding to A via this isomorphism.
• II) Find e u ∈ H ⊗ H (in general, this will be possible in closed form only if u (and, hence A) possesses additional structure).
• III) Find the matrix M corresponding to e u -this is e A .
Note: Throughout this work, tacit use of H ⊗ H representations of matrices in gl(4, R) will be made. These can be easily obtained from the entries of the 4 × 4 matrix in question (see [2,3,4] for some instances). In particular, R 4 = M j⊗i , J 4 = M 1⊗j .

Exponentials of Structured 4 × 4 Matrices
In this section the algebra isomorphism between H ⊗ H and real 4 × 4 matrices will be used to find exponentials of various structured matrices. For many of these matrices, their exponentials can be found directly from their H ⊗ H representations. These will be presented first. For the remaining the singular value factorization of matrices, no bigger than 3 × 3, is needed. This can be done in closed form, [8]. These will be presented next.

Exponentials Directly From H ⊗ H Representation
Below a (by no means exhaustive) list of nine families of real 4×4 matrices, whose exponentials can be directly found from their H ⊗ H representations, is presented. These families seem to be ubiquitous in applications.
For finding its exponential, it is noted that p ⊗ 1 and 1 ⊗ q commute, so the exponential of the sum is the product of the individual exponentials. Now consider Thus, p⊗1 is annihilated by a quadratic polynomial, and its exponential is therefore is Hence, e P is the matrix representation of {cosh( . Some Symmetric Toeplitz Matrices: The general case of a symmetric, Toeplitz matrix is subsumed by the case of bisymmetric matrices -see Remark (3.3) below.
Here we identify two important classes which do not require the intervening spectral factorization calculations for the general case.
• Symmetric, Toeplitz, Tridiagonal Matrix: Since such a matrix is met frequently in applications, it worth noting that its exponential can be directly computed in closed form. Indeed, their H ⊗ H representations are given by a(1 ⊗ 1) . Certain Normal Matrices: The general case of normal matrices is subsumed by the algorithm below for a symmetric matrix, since the case of skew-symmetric matrices has already been dealt with (a matrix is normal iff its symmetric and skew-symmetric parts commute). Here we discuss a subclass which does not require the spectral factorization calculations needed for exponentiating a symmetric matrix. This subclass is described via the following: Definition 3.1 Consider a normal N = S + T , with S its symmetric part and T its skew-symmetric part. Expressing T as the sum of two commuting skew-symmetric Such matrices will be called special normal.
Note that special normality forces T = 0. Special normality also implies that [S, T i ] = 0, i = 1, 2 (this will be shown below). It is this condition that makes exponentiation in closed form possible.
Indeed, consider, first the case that the assumption [S, T 1 ] = 0 forces, in conjunction with the linear independence of the elements e x ⊗ e y , e x , e y = i, j, k, each of the p ⊗ i, q ⊗ j, r ⊗ k to commute with s ⊗ 1.
This implies that each of p, q, r is parallel to s and hence the symmetric part of N can be expressed succinctly as Now the condition, [S, T 2 ] = 0 forces t to be parallel tot. Hence we find If T 1 = 0, then the condition [S, T 2 ] = 0 implies that each of p, q, r are parallel to one another (w.l.o.g p = 0), and hence S = p ⊗t, witht = kt, k ∈ R. Hence the above formula holds with minor modification.
Next, it will be shown that special normality implies [S, T i ] = 0, i = 1, 2. One first shows The calculation leading to the above simultaneously shows i) T 's minimal polynomial is quadratic iff either of s or t vanishes (in this case, trivially [S, Next, since S, T commute, they are simultaneously diagonalizable, via some unitary iff i) p is parallel to j and ii) q, r are perpendicular to j. If, in addition we suppose either q, r are parallel or q, r are perpendicular to one another, then exponentiation in closed form is possible. For brevity the former possibility is assumed. Hence Note, in particular, that RS is symmetric, persymmetric, but not Toeplitz.
Similarly, if JS = SJ, one finds (since J 2 = −I) that e JS = cos(S) + J 2n sin(S) Now if S, symmetric, commutes with J, then fortunately (or unfortunately) J 4 S is also simultaneously skew-symmetric, and therefore the previous formula is yet another way of exponentiating J 4 S. Hence, the details are omitted.

The General Symmetric Case
Exponentiating the general 4 × 4 symmetric matrix requires the spectral factorization of a 3 × 3 matrix (which can be done in closed form). Before getting to that, the principal enabling feature of the algorithm below is described by the following: Proof: It suffices to observes that each of the summands in a(1 ⊗ 1) commutes with each other due to the orthogonality property. The formula for e (ui⊗vi) is now just a consequence of (u i ⊗ v i ) squaring to a positive constant times the identity. will render the calculation of the exponential in closed form too. However, these will not be pursued here for brevity.

Remark 3.2
• i) Consider the element p ⊗ i + q ⊗ j + r ⊗ k, p, q, r ∈ P . Then, as observed Since the {u i }, {v i } are each an orthonormal triple, the exponential of p ⊗ i + q ⊗ j + r ⊗ k, which equals the exponential of 3 i=1 σ i u i ⊗ v i , can be explicitly found by using Proposition (3.1). The only issue is computing the singular value factorization of a real 3 × 3 matrix. However, this is the spectral factorization of a real 3 × 3 symmetric matrix, which itself can be done in closed form, [8]. It is interesting to note that the technique described in [8], consisting of 3 × 3 matrix manipulations, can itself be implemented via quaternions.
• ii) Note the subsequent rotations employed in [2] to diagonalize a symmetric matrix are not required, since diagonalization is not being employed here to compute exponentials.
Only the reduction to form used in Proposition (3.1) is needed.
This leads to the following algorithm for the exponential of a 4 × 4 symmetric matrix: • Represent the matrix as a(1 ⊗ 1) + p ⊗ i + q ⊗ j + r ⊗ k, p, q, r ∈ P .
• Compute the singular value factorization, • Compute the exponential of a(1 ⊗ 1) with α i , β i , γ i ∈ C. Now the fact that these constants are complex does not prevent each of X 1 , . . . , X 3 from commuting with each of X 4 , . . . , X 6 . Neither does it prevent each of X 1 , . . . , X 3 anti-commuting with one another nor each of X 4 , . . . , X 6 anti-commuting with one another. Finally, X 2 i = −c 2 i I 4 for each i = 1, . . . , 6, for some c i ∈ C. Hence the exponential is given by be the corresponding real 3 × 3 matrices. If these could be simultaneously brought into the canonical forms M i = 3 k=1 σ i k u k v T k , with the u k and v k orthonormal, σ i k ∈ R, then clearly the algorithm for real symmetric matrices would extend verbatim to such matrices. Many sufficient conditions are known for such simultaneous canonical form, [9]. One such condition

Conclusions
In this note, closed form formulae are provided for exponentials of several important families of real (and complex) 4×4 matrices. In conjunction, with techniques such as Givens rotations, these formulae provide algorithms for exponentiating classes of structured matrices in higher dimensions. The principal technique is the invocation of the associative algebra isomorphism between gl(4, R) and H ⊗ H. It is the ease of multiplication in H ⊗ H which facilitates the discovery of closed form exponentials for many matrices It is possible to write down exponentials of matrices once their minimal polynomial is known (especially if they are at most quartic). However, these formulae themselves can be quite complicated and hence they were not pursued in this note. This is exemplified by generic 4 × 4 skew-symmetric matrices, whose minimal polynomial is quartic. The corresponding exponential formula, though equivalent to the one given here, is substantially more complicated.
In Clearly,H ⊗H is associative algebra isomorphic to gl(4, c), whereH is the complexification of H. One can identify the latter with gl(2, C). However, it is better to view its elements This notion of conjugation is equivalent to Hermitian conjugation in gl(4, C). This does not, however, render calculating exponentials in su(4) as simple as in so(4, R) (after all one cannot run away from the curse of dimensionality by such an artifice). However, several Hermitian and skew-Hermitian matrices (e.g., whose real and imaginary parts come from special normal real matrices) are easily exponentiated.

Appendix I:
In this appendix, a different approach to the exponentiation of matrices in p(4, R), so(2, 2, R), p(3, R) is described, which reduces the problem to the exponentiation of 2×2 matrices (this is equally applicable to their complex counterparts). This is first illustrated for matrices in so(3, R) and so(4, R) since this should be reasonably well known terrain. Attention, in particular, is drawn to Remark (5.1), which provides the correct heurisitics needed to generalize this to the matrices in p(4, R), so(2, 2, R), p(3, R).
Consider an element A ∈ so(3, R). Its exponential can be computed explicitly via the Rodrigues formula. The usual derivation of this relies on the fact that A satisfies Any matrix which satisfies this equation will satisfy the Rodrigues formula. There is a equally well-known relation between su(2) and so(3, R). What is, perhaps, less appreciated is that this relation yields an explicit technique to find e A , A ∈ so(3, R). To describe this, fix a G ∈ SU (2). Consider V = {A | A * = A, Tr(A) = 0}. SU (2) acts via conjugation on elements A ∈ V , viz., φ G (A) = GAG −1 . It is well known, that upon identifying V with R 3 through the basis {σ k , k = x, y, z}, this action yields a proper rotation of R 3 . Thus, we get a homomorphism, φ : SU (2) → SO (3, R), which sends G to the matrix of φ G with respect to the basis {σ k , k = x, y, z}. This is a surjective, two-one, homomorphism. Linearizing this map, we get a Lie-algebra isomorphism ψ : su(2) → so(3, R), viz., ψ(A) is the matrix of the linear map which sends v ∈ V to Av − vA with respect to the {σ k , k = x, y, z} basis. with A ∈ su(2). This is a Lie-algebra isomorphism. From elementary considerations in Lie theory ψ and φ provide the following technique to find e A , A ∈ so(3, R): (2) • ii) Compute e B ∈ SU (2) -this can be explicitly done since satisfies the condition in i) of Remark (2.1)., This is arguably easier to use than the Rodrigues formula (it is left to the reader to verify that the two result in the same formula). This is not to disparage the Rodrigues formula -it applies to situations where Lie theory would have no visible role. But the fact that a 3 × 3 exponential has been computed with a 2 × 2 calculation is significant. Similar and even better savings occur by such arguments.
Exponentials in so(4, R): There is a well known two-one Lie group homomorphism denoted su(2) × su(2) → so(4, R) which sends (X, Y ) ∈ su(2) × su (2) to the matrix of the map (with respect to the I 2 , iσ k basis) which sends Z ∈ V to XZ − Y Z. This yields an algorithm to find e A , A ∈ so(4, R), which reduces to finding two 2 × 2 exponentials in su(2) -the statement of the algorithm is omitted (mimick the p(4, R) algorithm given below).
The corresponding relations between SL(2, C) (respectively SL(2, C) × SL(2, C)) and SO(3, C) (respectively SO(4, C)) once again reduce exponentiation of matrices in so (3, C) and so(4, C) to 2 × 2 calculations. Note that the fact that SO(3, C) etc., are not compact does not matter for the veracity of this procedure. All that is needed for finding e A is that the corresponding φ be a Lie group homomorphism (it need not even be surjective) and the corresponding ψ be a Lie algebra isomorphism.
Remark 5.1 Traditional proofs of the SU (2) covering of SO(3, R) proceed by i) using su(2) itself as the vector space V , and ii)then, by constructing a bilinear form, K(X, Y ) = Tr(ad X ad Y ) on su (2) and showing that this is preserved by the action of SU (2). For our purposes it is more useful to proceed differently. On any (sub)space of 2 × 2 matrices, there are two obvious candidates for quadratic forms, viz., i) Tr(X 2 ); and ii) det(X). One is even lead inexorably to these forms upon inspecting the forms of the maps φ used above for both so(3, R) and so(4, R). Polarizing these two leads to the following choices: It is easy to see that, with the choice of bases made in the derivation of the so(3, R) (resp. so(4, R)) algorithms, that the symmetric matrices representing these two forms are precisely 2I 3 (resp. I 4 ). This immediately shows that the matrix of the corresponding φ's are orthogonal. Linearizing φ, we get a technique to find exponentials in so(1, 3, R), cf., [7].
Below the same thinking is used to compute exponentials in p(4, R), so(2, 2, R) and p(3, R).
The method can be applied to several other Lie algebras stemming from symmetric, non-degenerate, bilinear forms on R 4 . However, we limit ourselves to these cases for brevity.

Exponentials in p(4, R):
Consider gl ( Algorithm for e A , A ∈ p(4, R): • i) Find the pair (g, h) = ψ −1 (A) ∈ sl(2, R) × sl(2, R); • ii) Find G = e g , H = e h . This is easily done since g, h satisfy the equation in Remark • iii) Find the matrix of φ G,H with respect to the above basis. This is e A .

Exponentials in p(3, R):
Now identify R 3 with the real span of E 12 , σ x , E 21 . This is sl(2, R). Then the matrix of L 1 (X, Y ), with respect to this basis, is, upto a constant, R 3 . Let SL(2, R) act on this copy of R 3 via φ G (h) = GhG −1 . This action preserves L 1 (X, Y ). Thus, the matrix of φ G is in P (3, R) and the map φ : SL(2, R) → P (3, R) is easily seen to be a Lie group homomorphism. Linearizing φ leads to a Lie algebra isomorphism ψ : sl(2, R) → p(3, R) which sends h ∈ sl(2, R) to the matrix of the linear map, which sends X ∈ sl(2, R) to hX − Xh (identifying sl(2, R) with R 3 via the above basis). This leads to an algorithm for finding e A , A ∈ p(3, R).

Remark 5.3 i)
The last calculation can be mimicked to find exponentials in so(2, 1, R).
Indeed, identify sl(2, R) with R 3 via the basis {σ x , σ z , E 12 − E 21 } and proceed verbatim as in the p(3, R) case. ii) All of the above calculations extend to find exponentials in p(4, C) etc., The only difference is one works with complexifications of the various Lie algebras introduced before, i.e., sl(2, C) × sl(2, C) for p(4, C) etc.,

Appendix II
In this appendix are listed i) five classes of matrices, each a Jordan algebra, which can be exponentiated by mimicking the technique for skew-Hamiltonian matrices; ii) eight classes of matrices, each forming a Lie algebra, which can be exponentiated by mimicking the technique for perskewsymmetric matrices. In most cases the technique extends to their complex analogues (e.g., so(2, 2, C)), cf., Remark (3.4). In both lists, both the H ⊗ H representation and the 2 × 2 block representations are provided. then one can find exponentials of matrices in J 2 (resp. L 2 ) provided G is explicitly described.
In the first list, the first two Jordan algebras pertain to bilinear forms which are equivalent to J 4 , while all the matrices in the second list stem from symmetric forms equivalent to R 4 .
While it is possible to explicitly construct the corresponding G's, it is far easier to work with the matrices in these lists directly.

Exponentials of Eight Lie Algebras
• so(2, 2, R). The H ⊗ H representation is a(1 ⊗ i) + p ⊗ i + b(i ⊗ i) + i ⊗ q, p, q ∈ span {j, k}, a, b ∈ R. The block representation is where B is any 2 × 2 matrix, while A, C are 2 × 2 anti-diagonal matrices with zero anti-trace.