Real logarithms of semi-simple matrices

We study the differential and topological structures of the set of real logarithms of any semi-simple non-singular matrix and of the set of real skew-symmetric logarithms of any special orthogonal matrix.


Introduction
The aim of this work is to study the differential geometric properties of the sets of real logarithms of real semi-simple matrices and of skew-symmetric real logarithms of special orthogonal matrices.As far as we know, such studies have never been done before.More generally, there are not many works that study the real logarithms of a matrix from a theoretical point of view.The most known is an old work by Culver ([2]), in which, among other things, the author proves that a non-singular real square matrix M has a real logarithm if and only if each of its Jordan blocks corresponding to a negative eigenvalue occurs an even number of times.Furthermore, Culver provides necessary and sufficient conditions, in terms of the Jordan blocks of M, for the real logarithm of M to be unique and for the This research was partially supported by GNSAGA-INdAM (Italy).
set of real logarithms of M to be countable.A simpler exposition of some of these findings can be found in [13].
After a first preliminary Section in which, in particular, the main homogeneous spaces involved in the structure theorems of the sets of real logarithms and of skew-symmetric real logarithms are defined, in Section 2 we determine, in the simplest cases, some of the homotopy groups of these homogeneous spaces.
The main result of Section 3 is Theorem 3.5, which states that the set of real logarithms of a given semi-simple matrix is a countable disjoint union of simply connected differentiable submanifolds of M (n, R), all diffeomorphic to suitable homogeneous spaces and whose dimensions depend on the eigenvalues of the matrices constituting each of them.This Theorem also states that the second homotopy group of each of these components is a free abelian group, whose rank can be expressed as a function of the eigenvalues of the matrices constituting the given component.
In Section 4 we prove a similar Theorem for the set of skew-symmetric real logarithms of a given special orthogonal matrix (see Theorem 4.4).In this case, each connected component of this set is a compact submanifold of so(n).
At the end of Sections 3 and 4, respectively, the sets of generalized principal real logarithms and of principal skew-symmetric real logarithms for suitable matrices are studied.The most interesting results we have obtained are stated in Theorems 3.8 and 4.7.Note that our Definitions 3.7 and 4.6 are more general than the usual definition of principal logarithm (see [8,Thm. 1.31 p. 20]).In fact, our principal logarithms are also defined for matrices with negative eigenvalues, even if, in general, they are not unique.For a study of the set of principal skew-symmetric real logarithms of a given special orthogonal matrix see also [4, §3].
We remark that the methods we use in this paper are very similar to those used in [5] to study the set of real square roots of suitable real matrices.
1. Preliminary facts 1.1.Notations.a) In this paper, for any integer n ≥ 1, we denote -M (n, R): the R-vector space of the real square matrices of order n; -GL(n, R) (and GL + (n, R)): the multiplicative group of the non-singular real matrices of order n (with positive determinant); -O(n) (and SO(n)): the multiplicative group of the real orthogonal matrices of order n (with determinant 1); so(n): the Lie algebra of the skew-symmetric real matrices of order n; -M (n, C): the C-vector space of the complex square matrices of order n; -GL(n, C): the multiplicative group of the non-singular complex matrices of order n; -U (n): the multiplicative group of the complex unitary matrices of order n; -I n : the identity matrix of order n; -O n : the null matrix of order n; i: the imaginary unit.
We write j X j to emphasize the union of mutually disjoint sets X j ; furthermore we denote by |S| the cardinality of any given finite set S and by δ (i,j) the usual Kronecker delta defined by δ (i,j) = 1 if i = j, and 0 otherwise.b) For every A ∈ M (n, C), tr(A) is its trace, A T is its transpose, A * := A T is its transpose conjugate, det(A) is its determinant and, provided that If C ∈ GL(n, R) , we denote by Λ C : X → CXC −1 the inner automorphism of GL(n, R) associated to the matrix C.
For every θ ∈ R, we denote It is easy to check that exp(δE) = E δ , for every δ ∈ R, and from this we get on its diagonal and, for every square matrix If S 1 , . . ., S m are sets of square matrices, then To give a full generality to the results of this paper (and to their proofs), it is necessary to establish agreements on the notations that we will use: if s is a non-negative integer parameter, whenever, in any formula, we write any term as       (• • • ), whenever the set I is empty.We also assign a meaning to the zero-order matrices I 0 , O 0 and to the zero-order groups SO(0), O(0), GL(0, R), GL + (0, R), defining them all equal to a single (phantom) point Q which, conventionally, satisfies the following conditions: for any set of complex square matrices S.
Moreover, we also agree that the zero-multiplicity eigenvalues of a given matrix X ∈ M (n, C) are all complex numbers that are not eigenvalues of X, while when we say that G is a free abelian group of rank zero, this means that G = {0}.
For all other notations and information on matrices, not explicitly recalled here, we refer to [9] and to [8].
Furthermore we have: ρ h (Z * ) = ρ h (Z) T , so the restriction of the monomorphism To simplify the notations and in absence of ambiguity, from now on we omit to write the symbol ρ h , so, for instance, we can consider M (h, C) as an R-subalgebra of M (2h, R), GL(h, C) as a Lie subgroup of GL(2h, R) and U (h) as a Lie subgroup of SO(2h).
b) For every matrix It is easy to prove that C B is a closed subgroup of GL(n, R).
If X ∈ M (n, R), we also denote by σ(X) the set of its distinct complex eigenvalues.Proof.Any matrix A ∈ GL(n, R) can be written in blocks as Fix i, j ∈ {1, • • • , t} with i = j.Since D j is semi-simple, there is a basis of C nj , say , consisting of eigenvectors of D j ; hence, if v is an element of this basis, then D j (v) = λv , for some λ ∈ σ(D j ), and so From the assumptions λ / ∈ σ(D i ), hence we conclude that A ij (v) = 0, for every A jj , where A jj ∈ GL(n j , R) and A jj D j = D j A jj , for j = 1, • • • , t .This concludes the proof.
Using elementary arguments similar to those of Lemma 1.3, it is easy to prove the following: , where 0 < θ < π and m ≥ 1.Then b) In the next Sections we will have to deal with the following homogeneous spaces: We also observe that Γ (0,ν) , Note that, if the matrix M ∈ GL + (n, R) is semi-simple, then it has at least one real logarithm if and only if each of its negative eigenvalues has even multiplicity.
1.8.Notations.Assume that M ∈ GL + (n, R) is semi-simple and that its (possible) negative eigenvalues have even multiplicity.
We want to study the following sets: Log(M ), the set of all real logarithms of M (see §3); the set of all skew-symmetric real logarithms of M , when M is supposed to be an element of SO(n) (see §4).
1.9.Remark.Let G be a real Lie group acting smoothly on a differentiable manifold X.The orbit of every x ∈ X is an immersed submanifold of X, diffeomorphic to the homogeneous space G G x , where G x is the isotropy subgroup of G at x.This submanifold is not necessarily embedded in X, but, if G is compact, then all orbits are embedded submanifolds (see [14]).

Some remarks on the homotopy groups of homogeneous spaces
As we will see, the homogeneous spaces we have defined in Remarks-Definitions 1.5 are involved in the study of the real logarithms of an arbitrary matrix.For this reason, in this section we will study some of their topological properties.We begin with some general properties concerning homogeneous spaces.
In this sequence the homotopy groups are based at the point e for G and H and at the point {e} for G/H ; the mappings ψ and ξ are, respectively, the homomorphisms induced by the natural inclusion H → G and by the projection on the quotient G → G/H, while the mappings δ are the connecting homomorphisms.
Proof.From the assumptions, it follows that the natural inclusion: G ′ → G is a bundle morphism, i.e. there exist a (natural) inclusion map: commutes.Then, for every i ≥ 2, we get the following commutative diagram, where the rows are exact sequences (see [16, p.91]): Here the maps f i , j i and l i are the homomorphisms induced by the natural inclusions.Since all groups are connected, if we define, as usual, , the previous commutative diagram also remains valid for i = 1.Furthermore, since G ′ and H ′ are deformation retracts of G and H, respectively, all the mappings f r , j r are isomorphisms, so that, by the classical Five-Lemma (see, for instance, [7, p. 129], in which the proof also works for non-abelian groups), the mappings l i are isomorphisms too , for every i ≥ 1.This concludes the proof of the Lemma.
the unique real symmetric logarithm of the positive definite matrix (XX T ) 1/2 , by Likewise, it is possible to prove that U (n) is a deformation retract of GL(n, C), GL(ν j , C) .Hence the Proposition follows from Lemma 2.2. and Θ (ν) reduce to a single point and so their homotopy groups are trivial.
We will study the other cases in the next Propositions of this Section.
Proof.The assumptions about ζ and s imply that is a cyclic group of order two.Furthermore, since π 2 (SO(ζ + 2ν)) = {0}, the final part of the exact homotopy sequence reduces to 0 In this sequence the homomorphism ψ is induced by the inclusion determined by the decomplexification mapping.Now we set we define, for j = 1, • • • , s, the following loops: . Hence, denoted by [α j ] and [β j ] the equivalence classes of the loops α j and β j in for every j = 1, • • • , s.The mapping ψ is surjective, since [β 1 ] is the generator (of order two) of π 1 (SO(ζ + 2ν)); so, by the exactness of the previous sequence, Moreover, all the loops β j are homotopic to the loop β 1 .Indeed, if Q j is a (special orthogonal) permutation matrix such that Q j β 1 (t)Q T j = β j (t) (for every t ∈ [0, 1]) and γ : [0, 1] → SO(ζ + 2ν) is a continuous path joining I (ζ+2ν) and Q j , then the mapping H defined by H(t, s) = γ(s)β 1 (t)γ(s) T (with t, s ∈ [0, 1]), is a homotopy between the loops β 1 and β j .Then ψ([ ) is a free abelian group of rank s and its generators are the homotopy classes of the loops α j for j = 1, • • • , s, so we have ψ( n j is even} is a free abelian group of rank ) → 0, for every i ≥ 3, and so (c) holds.
Proof.Taking into account that π 2 (U (ν)) = {0} and arguing as in Proposition 2.6, we obtain (a); consequently we have the following short exact sequence: Since π r (U (1)) = {0}, for every r ≥ 2, the exactness of the sequence The last claim of (c) follows from the well-known fact that π 3 (U (s)) ∼ = Z, for every s ≥ 2.

Real logarithms of semi-simple matrices
Given a real square matrix M of order n, we denote by Log(M ) the set of all real logarithms of M , namely Let M be semi-simple; in order for Log(M ) not to be empty, by Remark 1.7, we must assume that the matrix M is non-singular, with all (possible) negative eigenvalues having even multiplicity.
Aim of this section is to study Log(M ), for any M satisfying these assumptions.N i i! = 0 and this is impossible, since the degree of the minimal polynomial of N is k; so, we necessarily have k = 1, N = 0 and hence A is semi-simple.

3.2.
Remarks-Definitions. a) Let M ∈ GL + (n, R) be a semi-simple matrix, whose every (possible) negative eigenvalue has even multiplicity, and denote its eigenvalues in the following way: -the distinct positive eigenvalues are: -the distinct non-real eigenvalues are: where ρ (l,t) exp(±iθ l ) have both (positive) multiplicity m (l,t) , for every l, t , and

Note that
We also denote by 2A = 2 r l=1 a l the number of distinct non-real eigenvalues of M .
We have assumed that one or two of the indices p, r, q can be zero.For example, the index p vanishes when the matrix M has no positive eigenvalues.In this case, the numbers λ i , h i are not defined and it is understood that the term -ln(λ i ) ± 2πη (i,x) i , for x = 0, 1, • • • , b i , where we can assume , and where, if b i ≥ 1 and x = 1, • • • , b i , then the eigenvalues ln(λ i ) ± 2πη (i,x) i have both multiplicity u (i,x) ≥ 1, while, for x = 0, the multiplicity of ln(λ i ) is where τ (l,t,1) < • • • < τ (l,t,d (l,t) ) and In order to simplify notations and statements, we define the following sets: and the following multi-indices: If the set of multi-indices (η, u, τ, µ, σ, v) satisfies the conditions stated in (b), we say that it is admissible with respect to the matrix M or simply M-admissible.
Note that some multi-indices between η, u, τ, µ, σ, v are necessarily empty when p, r or q vanish.For instance, if p = 0, then η = u = ∅, and something like this when the integer r (or the integer q) is zero.
Note also that there is always a countable infinity of M -admissible sets of multiindices, unless the eigenvalues of M are all real, positive and simple, in which case there exists a single M -admissible set of multi-indices corresponding to the values We denote by L(M ) (η,τ,σ) the subset of Log(M ) of all real logarithms of M whose eigenvalues agree with the eigenvalues of the matrix Y (with the same multiplicities).We say that the eigenvalues of Y (each with its own multiplicity) are the eigenvalues (with corresponding multiplicity) of L(M ) (u,µ,v) (η,τ,σ) .Note also that, unless the eigenvalues of M are all real, positive and simple, we have where the countable disjoint union is taken on all M -admissible sets of multi-indices (η, u, τ, µ, σ, v), while, if the eigenvalues of M are all real, positive and simple, the set Log(M ) agrees with L(M ) (b), the real Jordan forms, J M of M and J of J, can be written, respectively, as follows: (−w j )I 2k j ; (⋆⋆) .
By [9,Cor. 3.4.1.10,p.203], we know that there exist two matrices Since J is a real Jordan form common to all matrices of L(M ) Taking into account Notations 1.1 (△), it is easy to check that we have exp( J ) = J M .Note also that this equality implies that C J ⊆ C JM .
for some X ∈ C JM .This gives one inclusion in the first equality of the statement.
The reverse inclusion is a simple computation.The second equality of the statement follows directly from the definition of the mapping Λ has the properties required for set L(M ) Let us consider the left action by conjugation of C JM on GL(n, R).The set {X J X −1 : X ∈ C JM } is the orbit of J .By Remark 1.9, this set is an immersed submanifold of GL(n, R), diffeomorphic to the homogeneous space is the isotropy subgroup of the action.
and this last set is closed and, therefore, it is an embedded submanifold of GL(n, R) (see for instance [12, § 2.13 Theorem, p. 65]).
3.4.Lemma.Let J M and J the matrices of Remarks-Definitions 3.2 (d) defined by (⋆) and (⋆⋆), respectively.Then the Lie groups of non-singular matrices commuting with J M and J are the following: Proof.The statement follows directly from Lemmas 1.3 and 1.4.
Denote by C an arbitrary connected component of L(M ) If, in addition, the multiplicity of all real eigenvalues of L(M ) (η,τ,σ) (u,µ,v) is less than or equal to 2, then, for every α ≥ 3, π α (C) is isomorphic to the direct sum Proof.From Proposition 3.3 and from Lemma 3.4, it follows that L(M ) (η,τ,σ) (u,µ,v) is a manifold diffeomorphic to the following product of homogeneous spaces: For part (c), we note that the condition on the non-real eigenvalues of L(M ) (η,τ,σ) (u,µ,v) is equivalent to u (i,x) = µ (l,t,z) = v (j,y) = 1, for every possible choice of the indices i, x, l, t, z, j, y, so that, under this condition, we have (•) h i = g i + 2b i , m (l,t) = d (l,t) , k j = c j , for all possible indices i, l, t, j.We denote by PLog(M ) the set of all generalized principal real logarithms of M .
Of course this set can be empty, but this does not happen if the matrix M is non-singular, semi-simple and all its negative eigenvalues have even multiplicity.
3.8.Theorem.Let M ∈ GL + (n, R) be a semi-simple matrix, whose distinct negative eigenvalues are exactly q (where q ≥ 0) and have multiplicity 2k 1 , • • • , 2k q , respectively.If q ≥ 1, the set PLog(M ) is a manifold with 2 q connected components, each of which is diffeomorphic to the symmetric space q j=1 Γ (0;k j ) , while, if M has no negative eigenvalues, the set PLog(M ) is a single point .Moreover, if C is any connected component of PLog(M ), then C is simply connected and π 2 (C) is a free abelian group of rank B, where B is the number of distinct negative eigenvalues of M , whose multiplicity is is greater than or equal to 4.
Proof.Using the same notations as in Remarks-Definition 3.2, we have here O indicates any multi-index whose entries are all zero.So, by Theorem 3.5 (a), the manifold PLog(M ) has 2 q connected components, which are, if q ≥ 1, all diffeomorphic to q j=1 Γ (0;k j ) , while PLog(M ) consists of a single point, when q = 0; indeed Γ (h i ) and Θ (m (l,t) ) reduce to a single point, for all possible indices i, l, t.
The final part of the statement follows from Theorem 3.5 (b), taking into account that, in this case, the set I is empty, d (l,t) = c j = 1 for all possible indices l, t, j, L = {j : 1 ≤ j ≤ q , k j = 1} and B = q − |L| .

Skew-symmetric real logarithms of special orthogonal matrices
In this Section we assume n ≥ 2. 4.1.Notations.Let M ∈ SO(n).Since the eigenvalues of M have unitary modulus, the real Jordan form of M can be written as follows: ( * ) so the eigenvalues of M are: 1 with multiplicity h ≥ 0, exp(±iθ 1 ) both with multiplicity m 1 , • • • , up to exp(±iθ r ) both with multiplicity m r (m j ≥ 1 , for every j, if r > 0) and −1 with multiplicity 2k ≥ 0.
Note that, also in this case, the integers h, r, k can vanish; so we assume, in this Section, the same agreements stated in Notations 1.1 (b).
Note also that, if n is odd, then h is also odd and, in particular, 1 is necessarily an eigenvalue of M .
It is well known that there exists where the countable disjoint union is taken on all M -admissible sets of multi-indices (η, u, τ, µ, σ, v).As in Remarks-Definitions 3.2 (c), L so(n) (M ) ) is an open and closed topological subspace of Log so(n) (M ), and so we get that each connected as in Definitions 4.2 ( * * ).Then we have (u,µ,v) .We know that there exists C ∈ O(n) such that W = C J C T .By Notations 1.1 (△), we get exp( J ) = J M .Hence, since , one of these two sets is a submanifold of so(n) if and only if the other is too, and in this case they are diffeomorphic.Now consider the left action ψ, of the compact Lie group (by Remark 1.9).This concludes the proof.(u,µ,v) is a compact homogeneous submanifold of so(n), whose connected components are all diffeomorphic to the product ; it has two connected components if and only if either 1 is eigenvalue of M , −1 is not eigenvalue of M and 0 is not eigenvalue of L so(n) (M )

1. 3 .
Lemma.Let D = t j=1 D j ∈ M (n, R), where each D j is a semi-simple real square matrix of order n j (with t j=1 n j = n) and assume that σ(D j ) ∩ σ(D h ) = ∅, as soon as j = h.Then we have C D = t j=1 C Dj .

2. 1 .
Remark.Let G be a connected Lie group with identity e and let H be any connected Lie subgroup of G. Denoted by G/H the related homogeneous space and by {e} = H the equivalence class of e in the quotient G/H, it is known that we have the following exact homotopy sequence, induced by the fibration on the quotient (see [16, p.90]):

3. 1 .
Remark.Let A ∈ M (n, R).Then A is semi-simple if and only if exp(A) is semi-simple.One implication is trivial.For the other, assume that exp(A) is semi-simple.By the additive Jordan-Chevalley decomposition, there are a semi-simple matrix S and a nilpotent matrix N of index k ≥ 1, such that A = S + N with SN = N S (see for instance [10, § 4.2] and also [3]).Since N and S commute, we have exp(A) = exp(S)exp(N ) with exp(S) semi-simple and exp(N ) unipotent, so, from the uniqueness of the multiplicative Jordan-Chevalley decomposition, we get exp(A) = exp(S) and exp(N ) = I n .If k ≥ 2, from the latter equation we get k−1 i=1 the same type, does not appear in the previous or similar equalities, in accordance with Notations 1.1 (b).Analogous remarks hold when r or q are zero.b)Let M be as in (a) and denote by Y ∈ M (n, R) a real logarithm of M .By Remark 3.1, Y is semi-simple and its eigenvalues are (complex) logarithms of the eigenvalues of M .Hence there exist two finite sets, {η (i,x) , τ (l,t,z) , σ (j,y) } ⊂ Z and {u (i,x) , b i , µ (l,t,z) , d (l,t) , v (j,y) , c j } ⊂ N, such that the distinct eigenvalues of Y are precisely the following:
τ,σ) is an open and closed topological subspace of Log(M ) and, consequently, every connected component of L(M ) (u,µ,v) (η,τ,σ) is also a connected component of Log(M ).d) If the semi-simple matrices M and Y (and their eigenvalues) are as in (a) and

3. 5 .
Theorem.Let M ∈ GL + (n, R) be a semi-simple matrix, whose eigenvalues are as in Remarks-Definitions 3.2 (a).Fix any set (η, u, τ, µ, σ, v) of M -admissible multi-indices, denote by 2A the number of distinct non-real eigenvalues of M and define the sets I, J, J, K, L as in Remarks-Definitions 3.2 (c).Then a) L(M ) (j,y) , C) ; hence, by Remarks-Definitions 1.5 (d), we can easily get the statement (a).By (a) and Remark 2.8, we get that the component C is simply connected and the rank of the free abelian group π 2 (C) is i∈I

.
From this and from (b), we get the requested formula for the rank of π 2 (C) .If, in addition, the condition on the real eigenvalues holds, then we have g i ≤ 2, for i = 1, • • • , p, so the formula for π α (C) (α ≥ 3) follows from Propositions 2.3, 2.6 (c) and 2.7 (c), taking into account the equalities (•) andthat, if i / ∈ I then π α (SO(h i )) = π α (SO(g i )) = {0}, for every α ≥ 3. 3.6.Remark.Let M ∈ GL + (n,R) be a semi-simple matrix whose negative eigenvalues have even multiplicity.By Remarks-Definitions 3.2 (c), Theorem 3.5 (a) and Remarks-Definitions 1.5 (b), Log(M ) is a finite set if and only if the eigenvalues of M are all real, positive and simple, and in this case, it consists of a single point (see [2, Thm.2]).In all other cases, we have that the set Log(M ) is countably infinite if and only if every manifold L(M ) (η,τ,σ) (h,m,k) has zero dimension, and so, taking into account Theorem 3.5 (a) and Remarks-Definitions 1.5 (b),(c), we get that the set Log(M ) is countably infinite if and only if all eigenvalues of M are simple and no eigenvalue of M is negative, in accordance with [2, Cor.p. 1151].3.7.Definition.Let M ∈ M (n, R).We say that a matrix X ∈ M (n, R) is a generalized principal real logarithm of M , if exp(X) = M and every eigenvalue of X has imaginary part in [−π, π].This definition is more general than the usual one of principal real logarithm (see for instance [8, Thm.1.31 p. 20]).

see for instance [ 9 ,
Cor. 2.5.11p. 137]).4.2.Definitions.a) If V ∈ so(n) and exp(V ) = M , we say that V is a skewsymmetric real logarithm of M .We denote by Log so(n) (M ) = Log(M ) ∩ so(n) the set of skew-symmetric real logarithms of M .Now fix W ∈ Log so(n) (M ).Since the eigenvalues of W are complex logarithms of those of M as in Remarks-Definitions 3.2, there exist two finite sets,