Computational Aspects of Some Exponential Identities

Abstract

The notion of the exponential of a matrix is usually introduced in elementary textbooks on ordinary differential equations when solving a constant coefficients linear system, also providing some of its properties and in particular one that does not hold unless the involved matrices commute. Several problems arise indeed from this fundamental issue, and it is our purpose to review some of them in this work, namely: (i) is it possible to write the product of two exponential matrices as the exponential of a matrix? (ii) is it possible to “disentangle” the exponential of a sum of two matrices? (iii) how to write the solution of a time-dependent linear differential system as the exponential of a matrix? To address these problems the Baker–Campbell–Hausdorff series, the Zassenhaus formula and the Magnus expansion are formulated and efficiently computed, paying attention to their convergence. Finally, several applications are also considered.

Appendix

1.1 5.4 Lie Algebras

A Lie algebra is a vector space $\mathfrak {g}$ together with a map $[\cdot , \cdot ]$ from $\mathfrak {g} \times \mathfrak {g}$ into $\mathfrak {g}$ called Lie bracket, with the following properties:

• 1. $[\cdot , \cdot ]$ is bilinear.

• 2. $[X, Y] = -[Y,X]$ for all $X, Y \in \mathfrak {g}$.

• 3. $[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0$ for all $X, Y, Z \in \mathfrak {g}$.

Condition 2 is called skew symmetry and Condition 3 is the Jacobi identity. One should remark that $\mathfrak {g}$ can be any vector space and that the Lie bracket operation $[\cdot , \cdot ]$ can be any bilinear, skew-symmetric map that satisfies the Jacobi identity. Thus, in particular, the space of all $n \times n$ (real or complex) matrices is a Lie algebra with the Lie bracket defined as the commutator $[A,B] = A B - B A$.

Associated with any $X \in \mathfrak {g}$ we can define a linear map $\mathrm {ad}_X: \mathfrak {g} \longrightarrow \mathfrak {g}$ which acts according to

$$\begin{aligned} \mathrm{ad}_{X}Y=[X,Y], \qquad \mathrm{ad}_{X}^j Y = [X, \mathrm{ad}_{X}^{j-1} Y], \qquad \mathrm{ad}_{X}^0 Y=Y, \qquad j \in \mathbb {N} \end{aligned}$$

(90)

for all $Y\in \mathfrak {g}$. The “ad” operator allows one to express nested Lie brackets in an easy way. Thus, for instance, [X, [X, [X, Y]]] can be written as $\mathrm{ad}_X^3 Y$. Moreover, as a consequence of the Jacobi identity, one has the following properties:

• 1. $\mathrm{ad}_{[X,Y]} = \mathrm{ad}_X \mathrm{ad}_Y - \mathrm{ad}_Y \mathrm{ad}_X = [\mathrm{ad}_X, \mathrm{ad}_Y]$

• 2. $\mathrm{ad}_Z [X,Y] = [X, \mathrm{ad}_Z Y] + [\mathrm{ad}_Z X, Y]$.

For matrix Lie algebras one has the important relation (see e.g. [36])

$$ \mathrm{e}^X \, Y \ \mathrm{e}^{-X} = \mathrm{e}^{\mathrm{ad}_X} Y = \sum _{k=0}^{\infty } \frac{1}{k!} \mathrm{ad}_X^k Y, $$

so that

$$ \mathrm{e}^X \, \mathrm{e}^Y \, \mathrm{e}^{-X} = \mathrm{e}^Z, \qquad \text {with} \qquad Z = \mathrm{e}^{\mathrm{ad}_X} Y. $$

The derivative of the matrix exponential map also plays an important role in our treatment. Given a matrix $\varOmega (t)$, then [36]

$$\begin{aligned} \begin{aligned}&\frac{d }{dt} \exp (\varOmega (t)) = d \exp _{\varOmega (t)}(\varOmega '(t)) \ \exp (\varOmega (t)), \\&\frac{d }{dt} \exp (\varOmega (t)) = \exp (\varOmega (t)) \, d \exp _{-\varOmega (t)}(\varOmega '(t)), \end{aligned} \end{aligned}$$

where $d \exp _{\varOmega }(C)$ is defined by the (everywhere convergent) power series

$$ d \exp _{\varOmega }(C) = \sum _{k=0}^{\infty } \frac{1}{(k+1)!} \mathrm{ad}_{\varOmega }^k (C) \equiv \frac{\mathrm{e}^{\mathrm{ad}_{\varOmega }}-I}{\mathrm{ad}_{\varOmega }}(C). $$

If the eigenvalues of the linear operator $\mathrm{ad}_{\varOmega }$ are different from $2 m \pi i$ with $m \in \{\pm 1, \pm 2, \ldots \}$ then the operator $d \exp _{\varOmega }$ is invertible [11, 36] and

$$ d \exp _{\varOmega }^{-1}(C) = \frac{\mathrm{ad}_{\varOmega }}{\mathrm{e}^{\mathrm{ad}_{\varOmega }}-I}(C) = \sum _{k=0}^{\infty } \frac{B_k}{k!} \mathrm{ad}_{\varOmega }^k (C), $$

where $B_k$ are the Bernoulli numbers.

1.2 5.5 Free Lie Algebras and Hall–Viennot Bases

Very often it is necessary to carry out computations in a Lie algebra when no particular algebraic structure is assumed beyond what is common to all Lie algebras. It is in this context, in particular, where the notion of free Lie algebra plays a fundamental role. Given an arbitrary index set I (either finite or countably infinite), we can say that a Lie algebra $\mathfrak {g}$ is free over the set I if [49]

• 1. for every $i \in I$ there corresponds an element $X_i \in \mathfrak {g}$;

• 2. for any Lie algebra $\mathfrak {h}$ and any function $i \mapsto Y_i \in \mathfrak {h}$, there exists a unique Lie algebra homomorphism $\pi : \mathfrak {g} \rightarrow \mathfrak {h}$ satisfying $\pi (X_i) = Y_i$ for all $i \in I$.

If $\mathscr {T} = \{ X_i : i \in I \} \subset \mathfrak {g}$, then the algebra $\mathfrak {g}$ can be viewed as the set of all Lie brackets of $X_i$. In this sense, we can say that $\mathfrak {g}$ is the free Lie algebra generated by $\mathscr {T}$ and we denote $\mathfrak {g} = \mathscr {L}(X_1,X_2,\ldots )$. Elements of $\mathscr {L}(X_1,X_2,\ldots )$ are called Lie polynomials.

It is important to remark that $\mathfrak {g}$ is a universal object, and that computations in $\mathfrak {g}$ can be applied in any particular Lie algebra $\mathfrak {h}$ via the homomorphism $\pi$ [49], just by replacing each abstract element $X_i$ with the corresponding $Y_i$.

In practical calculations, it is useful to represent a free Lie algebra by means of a basis (in the vector space sense). There are several systematic procedures to construct such a basis. Here, for simplicity, we will consider the free Lie algebra generated by just two elements $\mathscr {T} = \{X,Y\}$, and the so-called Hall–Viennot bases. A set $\{E_i \, : \, i=1,2,3,\ldots \} \subset \mathscr {L}(X,Y)$ whose elements are of the form

$$\begin{aligned} E_1=X, \quad E_2=Y, \quad \text {and} \quad E_i = [E_{i'},E_{i''}] \quad i\ge 3, \end{aligned}$$

(91)

with some positive integers $i', i''<i$ ($i=3,4,\ldots$) is a Hall–Viennot basis if there exists a total order relation $\succ$ in the set of indices $\{1,2,3,\ldots \}$ such that $i\succ i''$ for all $i\ge 3$, and the map

$$\begin{aligned} d:\{3,4,\ldots \}\longrightarrow & {} \{(j,k) \in \mathbb {Z} \times \mathbb {Z} \ : \ j\succ k \succeq j''\}, \quad \end{aligned}$$

(92)

$$\begin{aligned} d(i)=(i',i'')&\end{aligned}$$

(93)

(with the convention $1''=2''=0$) is bijective.

In [57, 67], Hall–Viennot bases are indexed by a subset of words (a Hall set of words) on the alphabet $\{x,y\}$. Such Hall set of words $\{w_i\ : \ i\ge 1\}$ can be obtained by defining recursively $w_i$ as the concatenation $w_{i'}w_{i''}$ of the words $w_{i'}$ and $w_{i''}$, with $w_1=x$ and $w_2=y$. In particular, if the map (92) is constructed in such a way that the total order relation $\succ$ is the natural order relation in $\mathbb {Z}$, i.e., >, then the first elements of the Hall set of words $w_i$ associated to the indices $i=1,2,\dots ,14$ are x, y, yx, yxx, yxy, yxxx, yxxy, yxyy, yxxxx, yxxxy, yxxyy, yxyyy, yxxyx, yxyyx. In consequence, the corresponding elements of the basis in $\mathscr {L}(X,Y)$ are

$$\begin{aligned} \begin{aligned}&X, \; Y, \; [Y,X], \; [[Y,X],X], \; [[Y,X],Y], \; [[[Y,X],X],X], \; [[[Y,X],X],Y], \; [[[Y,X],Y],Y], \\&\; [[[[Y,X],X],X],X], \; [[[[Y,X],X],X],Y], \; [[[[Y,X],X],Y],Y], \; [[[[Y,X],Y],Y],Y], \\&\; [[[Y,X],X],[Y,X]], \; [[[Y,X],Y],[Y,X]]. \end{aligned} \end{aligned}$$

(94)

Notice that if the total order is chosen as < instead, it results in the classical Hall basis as presented in [15].

On the other hand, the Lyndon basis can be constructed as a Hall–Viennot basis by considering the order relation $\succ$ as follows: $i \succ j$ if, in lexicographical order (i.e., the order used when ordering words in the dictionary), the Hall word $w_i$ associated to i comes before than the Hall word $w_j$ associated to j. The Hall set of words $\{w_i\ : \ i\ge 1\}$ corresponding to the Lyndon basis is the set of Lyndon words, which can be defined as the set of words w on the alphabet $\{x,y\}$ satisfying that, for arbitrary decompositions of w as the concatenation $w=u v$ of two non-empty words u and v, the word w is smaller than v in lexicographical order [42, 67]. The Lyndon words for $i=1,2,\dots ,14$ are x, y, xy, xyy, xxy, xyyy, xxyy, xxxy, xyyyy, xxyxy, xyxyy, xxyyy, xxxyy, xxxxy and the corresponding (Lyndon) basis in $\mathscr {L}(X,Y)$ is formed by

$$\begin{aligned} \begin{aligned}&X, \; Y, \; [X,Y], \; [[X,Y],Y], \; [X,[X,Y]], \; [[[X,Y],Y],Y], \; [X,[[X,Y],Y]], \; [X,[X,[X,Y]]], \\&\; [[[[X,Y],Y],Y],Y], \; [[X,[X,Y]],[X,Y]], \; [[X,Y],[[X,Y],Y]], \; [X,[[[X,Y],Y],Y]], \\&\; [X,[X,[[X,Y],Y]]], \; [X,[X,[X,[X,Y]]]]. \end{aligned} \end{aligned}$$

(95)

It is possible to compute the dimension $c_n$ of the linear subspace in the free Lie algebra generated by all the independent Lie brackets of order n, denoted by $\mathscr {L}_n(X,Y)$. This number is provided by the so-called Witt’s formula [15, 45]:

$$\begin{aligned} c_n = \frac{1}{n} \sum _{d|n} \mu (d) 2^{n/d}, \end{aligned}$$

(96)

where the sum is over all (positive) divisors d of the degree n and $\mu (d)$ is the Möbius function , defined by the rule $\mu (1) = 1$, $\mu (d) = (-1)^k$ if d is the product of k distinct prime factors and $\mu (d) = 0$ otherwise [45]. For $n \le 12$ one has explicitly

n	1	2	3	4	5	6	7	8	9	10	11	12
$c_n$	1	1	2	3	6	9	18	30	56	99	186	335