One-Parameter Groups of Linear Transformations

Roytvarf, Alexander A.

doi:10.1007/978-0-8176-8406-8_10

One-Parameter Groups of Linear Transformations

Alexander A. Roytvarf²

Chapter
First Online: 15 November 2012

2468 Accesses

Abstract

A one-parameter linear group in a vector space L is a continuous map $ g:t\mapsto {g^t}=g(t) $ on the space of parameters ℝ, taking values in a space of the linear operators on L and having the following properties: (1) g ⁰ = E and (2) the sum of parameter values corresponds to the composition $ {g^{s+t }}={g^s}\circ {g^t} $. Thus, one-parameter linear groups in ℝ are the continuous solutions of the functional equations g(0) = 1, g(t + s) = g(t)g(s). As is well known, exponential functions satisfy these equations. After solving the first problems in this chapter, readers will find that the continuous solutions are all exponential and, thus, will extract the usual properties of exponential functions from these functional equations (which corresponds to a real historical development).

The second part of the chapter is devoted to proper generalizations for one-parameter matrix groups. Readers will find that all of them are given by the matrix exponential series, as in the one-dimensional case. However, a multidimensional picture looks much more complicated. Working on these problems will familiarize readers with matrix analysis. In addition, readers may need familiarity with some tools used by number theory and the theory of differential equations, such as Euclid’s algorithm, Chinese remainder theorem, and Poincaré’s recurrence theorem; we introduce and discuss these tools.

In the third part of the chapter readers will find some elementary classic applications of one-parameter matrix groups in differential equation theory (Liouville formula), in complex analysis (e.g., complex exponential functions, Euler’s formula), for finite-dimensional functional spaces (spaces of quasipolynomials), and others. Readers will see how to deal with these problems using elementary tools of analysis and linear algebra.

Further, part of this chapter contains problems for readers possessing more advanced preliminary experience (although we introduce and discuss the necessary tools). While working on these problems, readers will come across many interesting things regarding analysis and linear algebra and become acquainted with important concepts (symplectic forms and others).

Readers will encounter far-reaching advances and applications of the subjects considered in the present chapters in powerful mathematical theories: of differential equations, Lie groups, and group representations (a guide to the literature is provided).

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Notes

1.
The fact that elements $ {g^t}-{g^{{{t_0}}}} $ in the space of linear operators GL(L) tend to 0 for t → t ₀ may be expressed in terms of by-coordinate tending to zero, that is, for all of (dimL)² coordinates with respect to any fixed basis in GL(L). However, readers may show that for a finite-dimensional L the continuity of a group is equivalent to continuities of the orbits: $ t\to {t_0}\Rightarrow {g^t}\to {g^{{{t_0}}}} $ if (and, evidently, only if) $ t\to {t_0}\Rightarrow {g^t}x\to {g^{{{t_0}}}}x,\forall x\in L $. (Proving a similar claim for infinite-dimensional spaces requires stipulating a kind of topological completeness of the space (the Banach-Steinhaus equicontinuity theorem). Interested readers may apply to Dunford and Schwartz (1957), Hille and Phillips (1957), Yosida (1965), Halmos (1967), Riesz and Sz.-Nagy (1972), Reed and Simon (1972), Rudin (1973), and references therein.)
2.
An ordinary autonomous differential equation, such that the solutions do not become infinite for a finite time, defines a one-parameter group of diffeomorphisms of the phase space, g ^t, called the phase flow: t $ \mapsto $ g ^t x (t ∈ ℝ) are the solutions of this equation under the initial conditions g ⁰ x = x. (The transformations g ^t are linear when the equation is linear.) A Lie group is a group and simultaneously a smooth manifold such that the group operations are smooth maps; in fact, it is a multiparameter, or multidimensional, group g ^t,u,…, but of course an identity such as $ {g^{{{t_1}+{t_2},{u_1}+{u_2},\ldots }}}={g^{{{t_1},{u_1},\ldots }}}{g^{{{t_2},{u_2},\ldots }}} $ may hold only for Abelian (commutative) Lie groups. Note that discrete, e.g., finite Hausdorff, groups are Lie groups. (What are their dimensions?) In the theory of group representations, abstract groups and functions on them are studied via homomorphisms (representations) of those groups onto groups of linear transformations of vector spaces. In this way various group-theoretic problems are reduced to problems in linear algebra; this is also very important for applications, e.g., when dealing with symmetry groups of physical systems. A detailed discussion of the contents of all these mathematical theories is beyond the scope of this problem book; and interested readers may consult Arnol’d (1975, 1978, 1989), Weyl (1939, 1952), Chevalley (1946–1955), Lang (1965), Vilenkin (1965), Nomizu (1956), Serre (1965), Adams (1969), Weil (1940), Bredon (1972), Kirillov (1972), Humphreys (1973), Naymark (1976), Helgason (1962), Kobayashi and Nomizu (1963, 1969), Bourbaki (1968–1982), Warner (1983), Fulton and Harris (1991), Alperin and Bell (1995), and references therein; in addition, readers may turn to the vast literature devoted to applications of group theory to differential geometry, functional analysis, and theoretical physics (theory of relativity, quantum mechanics, theory of solids, and other branches).
3.
The arithmetical nature of this constant has not been investigated. Specifically, it is not known if it is a rational number or not (1984) (Vinogradov 1977–1984).
4.
The necessary and sufficient conditions for inclusion of an invertible linear operator in a one-parameter linear group are discussed in section P10.29^** below. Here, a special kind of these groups is considered.
5.
In ergodicity theory it is proved that the trajectory of a uniform motion t $ \mapsto $ (ω ₁ t,…, ω _n t) along an n-dimensional torus ℝⁿ mod 2π = {(θ ₁,…, θ _n)} is dense everywhere when ω _i are linearly independent over the integers. This shows that the topology of a subgroup g(ℝ) ≅ $ \langle $ℝ,+$ \rangle $ within the group GL(L) can be very complex. It is very different from the topology of a linear subspace within a vector space.
6.
Readers not familiar with this fact may prove it by themselves or turn for the discussion to section H6.9 (in the “Polar and Singular Value Decomposition Theorems” problem group above).
7.
For this, readers should verify absolute (hence, commutative) convergences of those series, and the equalities of cosine and sine to the sums of their Taylor series, which might be done as follows. Prove that a function f(t) infinitely differentiable for ∣t − t ₀∣ ≤ r has its Taylor series at t ₀ absolutely convergent and is equal to the sum of this series at any point of that interval if the derivatives all together are uniformly bounded on any segment ∣t − t ₀∣ ≤ r ₀ < r: ∀r ₀ ∈ (0,r), that is, ∃C = C(r ₀) such that ∣f ^[n](t)∣ ≤ C for ∣t − t ₀∣ ≤ r ₀ and n = 0,1,… [actually, if ∀r ₀ ∈ (0,r), then f ^[n](t) ⋅ r ₀ ⁿ/n! = o(1) as n → ∞, uniformly on ∣t − t ₀∣ ≤ r ₀].
8.
In fact, exp(o(n)) lies in a connected component of E, which is a special orthogonal group $ SO(n)=\left\{ {B\in O(n):\det B=1} \right\} $ consisting of orientation-preserving matrices.
9.
The equivalence of (1) and (2) was discussed in section H8.11 related to the problem group “A Property of Orthogonal Matrices” above.
10.
Alternatively, it may be defined via an identity $ \langle $ x, Ay $ \rangle $ = $ \langle $ A ^* x, y $ \rangle $, ∀x,y, which in general is not equivalent unless the bilinear form is either symmetric or skew-symmetric.
11.
Geometrically, the value of the form [,] on a pair of vectors equals the oriented area of a parallelogram spanned on these vectors. Using the symbolism of skew-symmetry algebra (as discussed in the problem group “A Property of Orthogonal Matrices” above), [,] = x∧y where linear functionals x, y are Cartesian coordinates.
12.
Geometrically, [,] is a sum of the oriented area elements on two-dimensional planes where planes in any pair are mutually orthogonal. These planes are invariant with respect to realification of the multiplication by i: [,] = ∑x _i∧y _i.
13.
This group is not the direct product D × G, as, for example, D ∩ O(n) = {E,−E} ≅ ℤ(2) [for even n, D ∩ SO(n) = {E,−E}] and D ∩ U(n) ≅ $ {\mathbb S} $ ¹.
14.
Readers familiar with exact sequences and commutative diagrams may interpret this statement as follows: a diagram with rows
$$ \begin{array}{*{20}{c}} {\{0\}} & \to & {\mathfrak{sl}(n,\mathbb{R})} & \to & {\mathfrak{gl}(n,\mathbb{R})} & {\mathop{\to}\limits^{tr }} & \mathbb{R} & \to & {\{0\}} \\ {} & {} & {\downarrow \exp } & {} & {\downarrow \exp } & {} & {\downarrow \exp } & {} & {} \\ {\{E\}} & \to & {SL(n,\mathbb{R})} & \to & {GL(n,\mathbb{R})} & {\mathop{\to}\limits^{\det }} & {{{\mathbb{R}}^{+* }}} & \to & {\{1\}} \\ \end{array} $$
is commutative. (Draw a similar diagram for the case of a complex field.) In Lie group theory this and similar diagrams express the functoriality of an exponential map.
15.
And the powers B ^z = e ^{z log B} (z ∈ X), for complex matrices.
16.
ξ and η are referred to as commensurable if they have a common divisor, otherwise, as incommensurable.
17.
In reality, ancient scientists kept to geometric images, and dealt with line segments instead of real numbers. Interested readers will find extensive discussions of this and related topics in Van der Waerden (1950).
18.
In fact, ζ _n → 0 as n → ∞, but it is not necessary to use that because the incommensurability of ξ, η is already implied by the fact that the diminishing sequence of positive ζ _n is not a finite one.
19.
Actually, the boundedness follows from the finiteness of the volume and the convexity (Cassels 1957).
20.
Theorems 2–4 also belong to Hermann Minkowski.
21.
A special case of this theorem, the Chinese remainder theorem, states that for any set of integer numbers m ₁,…, m _n ≠ 0, where any pair is relatively prime, the indicated system of congruences is solvable for any r ₁,…,r _n..
22.
Recall that linear operations over finite-dimensional matrices are continuous.
23.
In fact, a series in a finite-dimensional space normally converges with respect to any norms if it does so with respect to one [because of the equivalence of the norms in a finite-dimensional space, as discussed in the “2 × 2 Matrices That Are Roots of Unity” problem group above (section H7.6)]. In turn, using the norm of maximum of absolute values of the coordinates with respect to a fixed basis shows that the normal convergence of any series in the finite-dimensional spaces implies convergence.
24.
Here, the desired formula follows from substituting A, B by sA, tB, respectively.
25.
Proving a similar claim for infinite-dimensional space requires stipulating a kind of topological completeness of the space (the Banach-Steinhaus equicontinuity theorem). Interested readers may consult Dunford and Schwartz (1957), Hille and Phillips (1957), Yosida (1965), Riesz and Sz.-Nagy (1972), Reed and Simon (1972), Rudin (1973), and references therein.
26.
Zero is considered a natural number.
27.
The coefficients in the remaining ε ^k are also traces of certain linear operators on certain vector spaces (connected to the source operator A); can the reader specify them? (Advanced readers familiar with skew symmetry, as discussed, for example, in the problem group “A Property of Orthogonal Matrices” above, will be able to do this.)
28.
Also, readers may do without a reference to the boundedness, arguing as follows. A monotonic function may have, at most, a countable number of discontinuities, which, if they exist, are finite jumps (why?); at the same time, by the claim in section P9.28, the function θ(t) would be discontinuous at all points if it were at some point. (Work out the details.)
29.
Similitude transformations A $ \mapsto $ C ⁻¹ ∘ A ∘ C with the orientation-preserving linear operators C transpose the elements inside the hyperboloids’ sheets, whereas transformations with the orientation-reversing operators C transpose the sheets of every hyperboloid.
30.
Any point of a hyperboloid becomes the vertex (corresponding to one of 2πnI, −2πnI) for a suitably defined scalar product in ℝ².
31.
This description is completely analogous to that in section P7.4 (the “2 × 2 Matrices That Are Roots of Unity” problem group above), with the difference that log E has a “logarithmic ramification,” whereas ${ n}\sqrt{E} $ has an “algebraic ramification.”
32.
In Lie group theory this is referred to as a left-invariant atlas.
33.
Recall that all linear maps of finite-dimensional spaces are continuous. Interested readers will find a far-reaching discussion concerning the map $ A\mapsto {A^{*}} $ in infinite-dimensional cases and related subjects in Riesz and Sz.-Nagy (1972) and references therein.
34.
Such a product is called semidirect.
35.
Also, readers can prove that for any two mutually transversal Lagrangian planes a third Lagrangian exists that is transversal to both of them.
36.
Actually, this is a restricted formulation because a coordinate system (p,v) [or (S,T)] may be defined only on open subsets of $ \mathfrak{M} $ projected onto this coordinate plane without singularities.
37.
A definition and basic properties of the Legendre transform were discussed previously in the problem group “Convexity and Related Classical Inequalities.”
38.
As discussed previously in (7), this subalgebra may be distinct from the whole algebra unless the source bilinear form is neither symmetric nor skew-symmetric.
39.
Origin-preserving isometries are always linear, at least in any normed spaces, as S. Mazur and S. Ulam proved; however, for non-Hilbert norms, this claim is not so easily obtained (Banach 1932 and references therein).
40.
A Jordan box of dimension zero is “no box.”
41.
The sign is “+” (“−”) when Θ preserves (resp. reverses) the usual orientation of the circle.
42.
Readers familiar with homotopies and coverings will recognize in this argument a special case of the rising path theorem, which makes it possible, in particular, to rise topological group structures and homomorphisms to a covering (Massey 1967; Sze-Tsen Hu 1959).
43.
The absolute value on a cyclic group, generated by this divisor, serves as the so-called Euclidean function, which makes this group a Euclidean module. Let M be a module over a ring R and ε a function on M taking values in a partially ordered set with the minimum condition. M is referred to as a Euclidean module, with Euclidean function ε, if division with a remainder can be performed in M:
$$ \forall \xi, \eta\ \in\ M:\ \ \ \ \xi\ \ne\ 0\ \ \ \ \Rightarrow\ \ \ \ \eta = k\xi + \zeta,\ \ \ with\ \ k\ \in\ R,\ \;\zeta = 0,\ \ or\ \ \varepsilon (\zeta ) < \varepsilon (\xi ). $$

Any set of elements of the Euclidean module has a GCD, that is, any element of a submodule generated by this set corresponding to a minimal value of ε is a GCD. For the Euclidean module, computations using Euclid’s algorithm are always finite and yield a GCD.
44.
The subspaces L _i are in a general position when codim ⋂L _i = ∑codim L _i for any finite collection of them. For example, the orthogonal complements to vectors of a Euclidean space that are linearly independent in their totality are in a general position. (Why?)
45.
So, the sequence 0 → ⋂ ℤm _i → ℤ → ∏ℤ(m _i) → 0 is exact (in the language of Homological algebra).
46.
In Abelian group theory, a cyclic group of order μ is usually denoted ℤ[μ].
47.
A nonnegative-element numerical series ∑a _n is a majorant for ∑A _n when ||A _n|| ≤ a _n, ∀n.
48.
Advanced readers are probably familiar with a generalization of this notion for manifolds and for infinite-dimensional spaces and manifolds. The differential of a function on an open set in an infinite-dimensional vector space is often referred to as the Frechét differential (or the Frechét derivative); for the manifolds, the term tangent map is commonly used. In the infinite-dimensional case, the definition of df(x ₀) includes the continuity requirement (which is automatically fulfilled for the finite-dimensional case).
49.
This claim allows infinite-dimensional generalizations. See Hörmander (1983) for details and connected results.
50.
Actually, the existence of n − 1 columns of this m × n Jacobi matrix in a neighborhood of (x ₀, f(x ₀)), their continuity in x ₀, plus at least the existence (without continuity) of the last column in (x ₀, f(x ₀)) will ensure the existence of df(x ₀).
51.
Therefore, with respect to any matrix norm (here and everywhere subsequently).
52.
Also, readers may establish a similar generalization for the series of multivariable vector/matrix functions; to do this, instead of the integration technique in section E10.5, a multivariable (infinite-variable) version of the intermediate value theorem should be applied. Interested readers will find a common scheme of such a proof in Dieudonné (1960).
53.
Also, readers may generalize Abel’s lemma (discussed previously in section H10.5) for the sum of a matrix power series ∑a _n X ⁿ as a function of X (replacing absolute convergence with normal convergence) and establish the infinite differentiability of this function for ||X|| < ||X ₀|| when a series $ \sum {{a_n}X_0^n} $ converges.
54.
By virtue of A, the terms’ order is unimportant.
55.
This claim may also be applied to a different method of proving the above equality e ^A+B = e ^A e ^B for commuting matrices (in both one- and multi-dimensional cases). Give such a proof.
56.
Direct using inequalities $ \ln n>\sum\limits_{k=2}^n {{k^{-1 }}} >\ln (n+1)-1 $ brings rougher estimates c _n ≤ 1 + ln n − ln (n + 1) (as 1 + ln n − ln (n + 1) ≥ 3/2 − 1/(n + 1) − ln 2; provide more details).
57.
Experienced readers probably are familiar with the Weierstrass theorem stating that the sum of a uniformly convergent series (in other terms, the limit of a uniformly convergent sequence) of continuous functions itself is continuous. It is not difficult to prove, but we do not refer to this theorem here.
58.
Readers familiar with elements of complex analysis know that the complex exponential function maps ℂ onto ℂ^*, analytically, but not one-to-one. Indeed, this map is not one-sheeted but periodic (or infinite-sheeted), having 2πiℤ as its group of periods. [This is also discussed in this problem group (section H10.24).] Or the global complex logarithmic map is infinite-valued. The ramification (branching) relates to a nonsimple connectedness of the domain ℂ^* = ℂ\{0}. This map cannot, due to its branching, be expanded into the Laurent series centered at the origin, although ℂ^* is an annulus, and actually, a one-valued (or univalent) branch exists in a simply connected domain obtained from ℂ by removing a ray of the vertex at the origin. In addition, the convergence radius of the Taylor series for (any branch of) the logarithmic map is finite with any center because it is equal to the distance from the center to the nearest singular point (which is the origin).
59.
There is a generalization stating that a subgroup of a direct sum of infinite cyclic groups is a similar direct sum itself. (The number of summands, called rank, is invariant, similar to the dimension of a vector space.) For finite ranks, this can be proved geometrically, generalizing the approach developed in section S10.13. D (iii). Also, it may be proved with purely algebraic methods using the following formulation: a subgroup of a free Abelian group is free itself. Interested readers will find such a proof in Lang (1965). A complete proof, considering infinite ranks, is in Fuchs (1970). Connected results, concerning free objects in categories of groups, modules, and others, are discussed in group theory and homological algebra.
60.
This is a ring for which any computation using Euclid’s algorithm is finite. More formally, it is defined as an integral domain considered as a Euclidean module (as defined in section E10.13. D) over itself whose Euclidean function has the following extra property: ab = c ⇒ ε (a), ε (b) ≤ ε (c). The Euclidean ring is a unitary one because the existence of the identity element is implied by the property of being a Euclidean module; indeed, if ε (a) = min(ε) and ae = a, then e is that element. (Fill in the details. Warning: the ring of even integers is a Euclidean module over ℤ, but it is not a Euclidean ring!) Also, the Euclidean ring is a principal ideal domain, but the unique factorization theorem (holding for any such domains) may be proved for Euclidean rings without use of an axiom of choice (using an induction on the ordering in the range of the Euclidean function). Interested readers will find the details in Van der Waerden (1971, 1967).
61.
A proof close to the original can be found in Fichtengoltz (1970).
62.
For example, defining a new order “≺” as 2 k ≺ 2n, 2 k + 1 ≺ 2n + 1 for k < n, 2 k + 1 ≺ 2n, ∀k,n, or even in a more sophisticated way, as $ \prod {p_i^{{{\alpha_i}}}} $ ≺ $ \prod {p_i^{{{\beta_i}}}} $, if, for some n, α _i ≤ β _i, ∀i > n and α _n < β _n. (In the last formula, the products are formally taken over all primes p ₁ = 2, p ₂ = 3,…, but actually each contains a finite number of factors with nonzero exponents.)
63.
Readers not experienced enough in ODE are encouraged to use a different method of establishing the linear independence of those quasimonomials (as functions on ℝ). Let some linear combination of these monomilas be identical to zero: $ f: = \sum {\left( {{p_j}(t)\cos {\omega_j}t+{q_j}(t)\sin {\omega_j}t} \right){e^{{{\lambda_j}t}}}} \equiv 0.$ The same arguments as were used in section E10.21 to prove Proposition 2 show that f cannot include nontrivial exponential or polynomial terms (because otherwise f would be unbounded; complete the details), so f has the form $ f=\sum {{A_j}\cos {\omega_j}t+{B_j}\sin {\omega_j}t} $, with constant coefficients A _j, B _j. We may assume that all ω _j ≥ 0. (Why?) We must verify that the coefficients vanish. (The summand B _j sin 0 corresponding to ω _j = 0 may be excluded from consideration.) The summands of f may be expanded into power series, convergent for any t, so f may be expanded into power series itself. Such a series is unique (and thus is a Taylor series) because a function that is the sum of a convergent power series cannot possess a nondiscrete zero set. (Fill in the details.) Hence, Taylor expansions of the summands of f are summed in a zero series, which, using commutative convergence (as discussed in section S10.18 above), yields the equations
$$ \sum {{A_j}} =0,\quad \sum {{A_j}\omega_j^2} =0,\quad \sum {{A_j}\omega_j^4} =0,\ldots \quad \mathrm{ and}\quad \sum {{B_j}} {\omega_j}=0,\quad \sum {{B_j}\omega_j^3} =0,\quad \sum {{B_j}\omega_j^5} =0,\ldots. $$

The generalized Vandermonde determinants $ \det \left( {\begin{array}{*{20}{c}} {\omega_0^{{{k_0}}}} & \cdots & {\omega_n^{{{k_0}}}} \\ \vdots & {} & \vdots \\ {\omega_0^{{{k_n}}}} & \cdots & {\omega_n^{{{k_n}}}} \\ \end{array}} \right) $ are positive for 0 < ω ₀ < … < ω _n and k ₀ < … < k _n. [Readers should try to prove these relations; also, we give a proof subsequently in the “Least Squares and Chebyshev Systems” problem group: a quite elementary proof, for a special case of natural k _j, is discussed in section H12.4, and in the general case, a proof follows from the results of section P12.10; in addition, a different proof can be found in Polya et al. (1964).] Therefore, we will find that A _j and B _j vanish. QED. (We leave it to the reader to fill in the details.)
64.
Readers familiar as well with coverings know that, topologically, an exponential map realizes an infinite-sheeted covering $ \left\{ {\begin{array}{*{20}{c}} \mathbb{C} & \cong & {\mathbb{R}\times \mathbb{R}} & \to & {{{\mathbb{R}}^{+* }}\times {{\mathbb{S}}^1}\cong {{\mathbb{C}}^{*}}} \\ z & = & {x+yi} & \mapsto & {{e^x}\cdot {e^{yi }}} \\ \end{array}} \right. $. The raising of a closed loop [0,2π] → ℂ^*, rotating n times around the origin, into the total space of the covering is an open loop connecting points u, v, of the kth and (k + n)th sheets of this covering, respectively, such that e ^u = e ^v. (Produce a figure.) Pay attention to the realization of the raising, as it is made by the logarithmic map (inverse to the exponential), defined by integrating the differential form dz/z along the source loop in ℂ^*!

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Rishon LeZion, Israel
Alexander A. Roytvarf

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Roytvarf, A.A. (2013). One-Parameter Groups of Linear Transformations. In: Thinking in Problems. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8406-8_10

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DOI: https://doi.org/10.1007/978-0-8176-8406-8_10
Published: 15 November 2012
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