4 Chapter Basics of Classical Lie Groups: The Exponential Map, Lie Groups, and Lie Algebras
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4.1
The Exponential Map
The inventors of Lie groups and Lie algebras (starting with Lie!) regarded Lie groups as groups of symmetries of various topological or geometric objects. Lie algebras were viewed as the innitesimal transformations associated with the symmetries in the Lie group.
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For example, the group SO(n) of rotations is the group of orientation-preserving isometries of the Euclidean space En. The Lie algebra so(n, R) consisting of real skew symmetric nn matrices is the corresponding set of innitesimal rotations. The geometric link between a Lie group and its Lie algebra is the fact that the Lie algebra can be viewed as the tangent space to the Lie group at the identity. There is a map from the tangent space to the Lie group, called the exponential map. The Lie algebra can be considered as a linearization of the Lie group (near the identity element), and the exponential map provides the delinearization, i.e., it takes us back to the Lie group.
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These concepts have a concrete realization in the case of groups of matrices, and for this reason we begin by studying the behavior of the exponential maps on matrices. We begin by dening the exponential map on matrices and proving some of its properties. The exponential map allows us to linearize certain algebraic properties of matrices. It also plays a crucial role in the theory of linear dierential equations with constant coecients. But most of all, as we mentioned earlier, it is a stepping stone to Lie groups and Lie algebras. The last section gives a quick introduction to Lie groups and Lie algebras.
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We dene manifolds as embedded submanifolds of RN , and we dene linear Lie groups, using the famous result of Cartan (apparently actually due to Von Neumann) that a closed subgroup of GL(n, R) is a manifold, and thus, a Lie group. This way, Lie algebras can be computed using tangent vectors to curves of the form t A(t), where A(t) is a matrix. Given an n n (real or complex) matrix A = (ai, j ), we would like to dene the exponential eA of A as the sum of the series Ap Ap A e = In + = , p! p! p1 p0 letting A0 = In. The following lemma shows that the above series is indeed absolutely convergent.
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Lemma 4.1.1 Let A = (ai j ) be a (real or complex) n n matrix, and let = max{|ai j | | 1 i, j n}. If Ap = (apj ), then i |apj | (n)p i for all i, j, 1 i, j n. As a consequence, the n2 series apj i p! p0 converge absolutely, and the matrix e =
p0 A
Ap p!
is a well-dened matrix.
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It is instructive to compute explicitly the exponential of some simple matrices. As an example, let us compute the exponential of the real skew symmetric matrix A= 0 0 .
We nd that e =
A
cos sin sin cos
.
Thus, eA is a rotation matrix! This is a general fact. If A is a skew symmetric matrix, then eA is an orthogonal matrix of determinant +1, i.e., a rotation matrix. Furthermore, every rotation matrix is of this form; i.e., the exponential map from the set of skew symmetric matrices to the set of rotation matrices is surjective.
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In order to prove these facts, we need to establish some properties of the exponential map. But before that, let us work out another example showing that the exponential map is not always surjective. Let us compute the exponential of a real 2 2 matrix with null trace of the form A= a b c a .
We need to nd an inductive formula expressing the powers An. Observe that A2 = (a2 + bc)I2 = det(A)I2. If a2 + bc = 0, we have eA = I2 + A. If a2 + bc < 0, let > 0 be such that 2 = (a2 + bc).
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Then, A2 = 2I2, and we get sin A. e = cos I2 +
A
If a2 + bc > 0, let > 0 be such that 2 = (a2 + bc). Then A2 = 2I2, and we get sinh A, where cosh = (e + e )/2 and sinh = (e e )/2. eA = cosh I2 + It immediately veried that in all cases, det eA = 1.
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This shows that the exponential map is a function from the set of 2 2 matrices with null trace to the set of 2 2 matrices with determinant 1. This function is not surjective. Indeed, tr(eA) = 2 cos when a2 + bc < 0, tr(eA) = 2 cosh when a2 + bc > 0, and tr(eA) = 2 when a2 + bc = 0. As a consequence, for any matrix A with null trace, tr eA 2, and any matrix B with determinant 1 and whose trace is less than 2 is not the exponential eA of any matrix A with null trace. For example, B= a 0 0 a1 ,
where a < 0 and a = 1, is not the exponential of any matrix A with null trace.
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A fundamental property of the exponential map is that if 1, . . . , n are the eigenvalues of A, then the eigenvalues of eA are e1 , . . . , en . For this we need two lemmas. Lemma 4.1.2 Let A and U be (real or complex) matrices, and assume that U is invertible. Then e
U AU 1
= U eAU 1.
Say that a square matrix A is an upper triangular matrix if it has the following shape, a1 1 a1 2 a1 3 . . . a1 n1 a1 n 0 a2 2 a2 3 . . . a2 n1 a2 n 0 0 a3 3 . . . a3 n1 a3 n . . . . . , ... . . . . . 0 0 0 . . . an1 n1 an1 n 0 0 0 ... 0 an n i.e., ai j = 0 whenever j < i, 1 i, j n.
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Lemma 4.1.3 Given any complex n n matrix A, there is an invertible matrix P and an upper triangular matrix T such that A = P T P 1. Remark: If E is a Hermitian space, the proof of Lemma 4.1.3 can be easily adapted to prove that there is an orthonormal basis (u1, . . . , un) with respect to which the matrix of f is upper triangular. In terms of matrices, this means that there is a unitary matrix U and an upper triangular matrix T such that A = U T U . This is usually known as Schurs lemma. Using this result, we can immediately rederive the fact that if A is a Hermitian matrix, then there is a unitary matrix U and a real diagonal matrix D such that A = U DU . If A = P T P 1 where T is upper triangular, note that the diagonal entries on T are the eigenvalues 1, . . . , n of A.
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Indeed, A and T have the same characteristic polynomial. This is because if A and B are any two matrices such that A = P BP 1, then det(A I) = det(P BP 1 P IP 1), = det(P (B I)P 1), = det(P ) det(B I) det(P 1), = det(P ) det(B I) det(P )1, = det(B I). Furthermore, it is well known that the determinant of a matrix of the form 1 a1 2 a1 3 ... a1 n1 a1 n 0 2 a2 3 ... a2 n1 a2 n 0 0 3 . . . a3 n1 a3 n . . . . . ... . . . . . 0 0 0 . . . n1 an1 n 0 0 0 ... 0 n is (1 ) (n ).
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Thus the eigenvalues of A = P T P 1 are the diagonal entries of T . We use this property to prove the following lemma: Lemma 4.1.4 Given any complex n n matrix A, if 1, . . . , n are the eigenvalues of A, then e1 , . . . , en are the eigenvalues of eA. Furthermore, if u is an eigenvector of A for i, then u is an eigenvector of eA for ei . As a consequence, we can show that det(eA) = etr(A), where tr(A) is the trace of A, i.e., the sum a1 1 + +an n of its diagonal entries, which is also equal to the sum of the eigenvalues of A.
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This is because the determinant of a matrix is equal to the product of its eigenvalues, and if 1, . . . , n are the eigenvalues of A, then by Lemma 4.1.4, e1 , . . . , en are the eigenvalues of eA, and thus det eA = e1 en = e1++n = etr(A). This shows that eA is always an invertible matrix, since ez is never null for every z C. In fact, the inverse of eA is eA, but we need to prove another lemma. This is because it is generally not true that eA+B = eAeB , unless A and B commute, i.e., AB = BA. Lemma 4.1.5 Given any two complex n n matrices A, B, if AB = BA, then eA+B = eAeB .
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Now, using Lemma 4.1.5, since A and A commute, we have eAeA = eA+A = e0n = In, which shows that the inverse of eA is eA. We will now use the properties of the exponential that we have just established to show how various matrices can be represented as exponentials of other matrices.
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4.2
The Lie Groups GL(n, R), SL(n, R), O(n), SO(n), the Lie Algebras gl(n, R), sl(n, R), o(n), so(n), and the Exponential Map
The set of real invertible n n matrices forms a group under multiplication, denoted by GL(n, R). The subset of GL(n, R) consisting of those matrices having determinant +1 is a subgroup of GL(n, R), denoted by SL(n, R). It is also easy to check that the set of real nn orthogonal matrices forms a group under multiplication, denoted by O(n). The subset of O(n) consisting of those matrices having determinant +1 is a subgroup of O(n), denoted by SO(n). We will also call matrices in SO(n) rotation matrices.
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Staying with easy things, we can check that the set of real n n matrices with null trace forms a vector space under addition, and similarly for the set of skew symmetric matrices. Denition 4.2.1 The group GL(n, R) is called the general linear group, and its subgroup SL(n, R) is called the special linear group. The group O(n) of orthogonal matrices is called the orthogonal group, and its subgroup SO(n) is called the special orthogonal group (or group of rotations). The vector space of real n n matrices with null trace is denoted by sl(n, R), and the vector space of real n n skew symmetric matrices is denoted by so(n). Remark: The notation sl(n, R) and so(n) is rather strange and deserves some explanation. The groups GL(n, R), SL(n, R), O(n), and SO(n) are more than just groups.
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They are also topological groups, which means that they 2 are topological spaces (viewed as subspaces of Rn ) and that the multiplication and the inverse operations are continuous (in fact, smooth). Furthermore, they are smooth real manifolds. The real vector spaces sl(n) and so(n) are what is called Lie algebras. However, we have not dened the algebra structure on sl(n, R) and so(n) yet. The algebra structure is given by what is called the Lie bracket, which is dened as [A, B] = AB BA. Lie algebras are associated with Lie groups.
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What is going on is that the Lie algebra of a Lie group is its tangent space at the identity, i.e., the space of all tangent vectors at the identity (in this case, In). In some sense, the Lie algebra achieves a linearization of the Lie group. The exponential map is a map from the Lie algebra to the Lie group, for example, exp: so(n) SO(n) and exp: sl(n, R) SL(n, R). The exponential map often allows a parametrization of the Lie group elements by simpler objects, the Lie algebra elements.
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One might ask, what happened to the Lie algebras gl(n, R) and o(n) associated with the Lie groups GL(n, R) and O(n)? We will see later that gl(n, R) is the set of all real n n matrices, and that o(n) = so(n). The properties of the exponential map play an important role in studying a Lie group. For example, it is clear that the map exp: gl(n, R) GL(n, R) is well-dened, but since every matrix of the form eA has a positive determinant, exp is not surjective.
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Similarly, since det(eA) = etr(A), the map exp: sl(n, R) SL(n, R) is well-dened. However, we showed in Section 4.1 that it is not surjective either. As we will see in the next theorem, the map exp: so(n) SO(n) is well-dened and surjective. The map exp: o(n) O(n) is well-dened, but it is not surjective, since there are matrices in O(n) with determinant 1. Remark: The situation for matrices over the eld C of complex numbers is quite dierent, as we will see later.
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We now show the fundamental relationship between SO(n) and so(n). Theorem 4.2.2 The exponential map exp: so(n) SO(n) is well-dened and surjective. When n = 3 (and A is skew symmetric), it is possible to work out an explicit formula for eA. For any 3 3 real skew symmetric matrix 0 c b 0 a , A= c b a 0 letting = a2 + b2 + c2 and 2 a ab ac B = ab b2 bc , ac bc c2 we have the following result known as Rodriguess formula (1840):
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Lemma 4.2.3 The exponential map exp: so(3) SO(3) is given by eA = cos I3 + or, equivalently, by sin (1 cos ) 2 A+ A 2 if = 0, with e03 = I3. eA = I3 + The above formulae are the well-known formulae expressing a rotation of axis specied by the vector (a, b, c) and angle . Since the exponential is surjective, it is possible to write down an explicit formula for its inverse (but it is a multivalued function!). This has applications in kinematics, robotics, and motion interpolation. (1 cos ) sin B, A+ 2
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4.3
Symmetric Matrices, Symmetric Positive Denite Matrices, and the Exponential Map
Recall that a real symmetric matrix is called positive (or positive semidenite) if its eigenvalues are all positive or null, and positive denite if its eigenvalues are all strictly positive. We denote the vector space of real symmetric n n matrices by S(n), the set of symmetric positive matrices by SP(n), and the set of symmetric positive denite matrices by SPD(n). The next lemma shows that every symmetric positive denite matrix A is of the form eB for some unique symmetric matrix B. The set of symmetric matrices is a vector space, but it is not a Lie algebra because the Lie bracket [A, B] is not symmetric unless A and B commute, and the set of symmetric (positive) denite matrices is not a multiplicative group, so this result is of a dierent avor as Theorem 4.2.2.
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Lemma 4.3.1 For every symmetric matrix B, the matrix eB is symmetric positive denite. For every symmetric positive denite matrix A, there is a unique symmetric matrix B such that A = eB . Lemma 4.3.1 can be reformulated as stating that the map exp: S(n) SPD(n) is a bijection. It can be shown that it is a homeomorphism. In the case of invertible matrices, the polar form theorem can be reformulated as stating that there is a bijection between the topological space GL(n, R) of real n n invertible matrices (also a group) and O(n) SPD(n). As a corollary of the polar form theorem (Theorem 2.2.1) and Lemma 4.3.1, we have the following result:
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For every invertible matrix A there is a unique orthogonal matrix R and a unique symmetric matrix S such that A = R eS . Thus, we have a bijection between GL(n, R) and O(n) S(n). But S(n) itself is isomorphic to Rn(n+1)/2. Thus, there is a bijection between GL(n, R) and O(n) Rn(n+1)/2. It can also be shown that this bijection is a homeomorphism. This is an interesting fact. Indeed, this homeomorphism essentially reduces the study of the topology of GL(n, R) to the study of the topology of O(n). This is nice, since it can be shown that O(n) is compact.
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In A = R eS , if det(A) > 0, then R must be a rotation matrix (i.e., det(R) = +1), since det eS > 0. In particular, if A SL(n, R), since det(A) = det(R) = +1, the symmetric matrix S must have a null trace, i.e., S S(n) sl(n, R). Thus, we have a bijection between SL(n, R) and SO(n) (S(n) sl(n, R)). We can also use the results of Section 1.3 to show that the exponential map is a surjective map from the skew Hermitian matrices to the unitary matrices.
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4.4
The Lie Groups GL(n, C), SL(n, C), U(n), SU(n), the Lie Algebras gl(n, C), sl(n, C), u(n), su(n), and the Exponential Map
The set of complex invertible nn matrices forms a group under multiplication, denoted by GL(n, C). The subset of GL(n, C) consisting of those matrices having determinant +1 is a subgroup of GL(n, C), denoted by SL(n, C). It is also easy to check that the set of complex n n unitary matrices forms a group under multiplication, denoted by U(n). The subset of U(n) consisting of those matrices having determinant +1 is a subgroup of U(n), denoted by SU(n).
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We can also check that the set of complex n n matrices with null trace forms a real vector space under addition, and similarly for the set of skew Hermitian matrices and the set of skew Hermitian matrices with null trace. Denition 4.4.1 The group GL(n, C) is called the general linear group, and its subgroup SL(n, C) is called the special linear group. The group U(n) of unitary matrices is called the unitary group, and its subgroup SU(n) is called the special unitary group. The real vector space of complex n n matrices with null trace is denoted by sl(n, C), the real vector space of skew Hermitian matrices is denoted by u(n), and the real vector space u(n) sl(n, C) is denoted by su(n).
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Remarks: (1) As in the real case, the groups GL(n, C), SL(n, C), U(n), and SU(n) are also topological groups (viewed 2n2 as subspaces of R ), and in fact, smooth real manifolds. Such objects are called (real) Lie groups. The real vector spaces sl(n, C), u(n), and su(n) are Lie algebras associated with SL(n, C), U(n), and SU(n). The algebra structure is given by the Lie bracket, which is dened as [A, B] = AB BA. (2) It is also possible to dene complex Lie groups, which means that they are topological groups and smooth complex manifolds. It turns out that GL(n, C) and SL(n, C) are complex manifolds, but not U(n) and SU(n).
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One should be very careful to observe that even though the Lie algebras sl(n, C), u(n), and su(n) consist of matrices with complex coecients, we view them as real vector spaces. The Lie algebra sl(n, C) is also a complex vector space, but u(n) and su(n) are not! Indeed, if A is a skew Hermitian matrix, iA is not skew Hermitian, but Hermitian! Again the Lie algebra achieves a linearization of the Lie group. In the complex case, the Lie algebras gl(n, C) is the set of all complex n n matrices, but u(n) = su(n), because a skew Hermitian matrix does not necessarily have a null trace.
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The properties of the exponential map also play an important role in studying complex Lie groups. For example, it is clear that the map exp: gl(n, C) GL(n, C) is well-dened, but this time, it is surjective! One way to prove this is to use the Jordan normal form. Similarly, since det eA = etr(A), the map exp: sl(n, C) SL(n, C) is well-dened, but it is not surjective! As we will see in the next theorem, the maps exp: u(n) U(n) and exp: su(n) SU(n) are well-dened and surjective.
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Theorem 4.4.2 The exponential maps exp: u(n) U(n) and exp: su(n) SU(n)
are well-dened and surjective. We now extend the result of Section 4.3 to Hermitian matrices.
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4.5
Hermitian Matrices, Hermitian Positive Denite Matrices, and the Exponential Map
Recall that a Hermitian matrix is called positive (or positive semidenite) if its eigenvalues are all positive or null, and positive...