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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Lie Group Methods for Eigenvalue Function

Nazarkandi HA*

Departement of Mathematics, Marand Branch, Islamic Azad University, Marand, Iran

*Corresponding Author:
Nazarkandi HA
Departement of Mathematics
Marand Branch, Islamic Azad University
Marand, Iran
E-mail: [email protected]

Received Date: February 13, 2016; Accepted Date: June 15, 2016; Published Date: June 20, 2016

Citation: Nazarkandi HA (2016) Lie Group Methods for Eigenvalue Function. J Generalized Lie Theory Appl 10:240. doi:10.4172/10.4172/1736-4337.1000240

Copyright: © 2016 Nazarkandi HA. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

By considering a C∞ structure on the ordered non-increasing of elements of Rn, we show that it is a differentiable manifold. By using of Lie groups, we show that eigenvalue function is a submersion. This fact is used to prove some results. These results is applied to prove a few facts about spectral manifolds and spectral functions. Orthogonal matrices act on the real symmetric matrices as a Lie transformation group. This fact, also, is used to prove the results.

Keywords

Lie group; Spectral manifold; Submersion; Eigenvalue function; Spectral function

Preliminary

Let’s S(n, R) be the space of real symmetric matrices and O(n,R) the group of all real orthogonal matrices. For any A∈S(n,R) its (repeated) eigenvalues image are real, and it admits a spectral decomposition of the form

image (1.0)

(Where A is a symmetric matrix, P is an orthogonal matrix (PP^t=I),\lambda_1,…\lambda_n are eigenvalues of A.)

for some P∈O(n,R), where diag[λ1,…,λn] is the diagonal matrix with its ith diagonal entry λi and Pt is the transpose of matrix P. Note that (1.0) is independent of the choice of P∈O(n,R) [1,2].

Let λ(.):S(n,R)→Rn be the eigenvalue function such that λi(A),i=1,… ,n, yield eigenvalues of A for any A∈S(n,R) and are ordered in a nonincreasing order, that is, λ1(A)≥…≥λn(A). For applications of this function in variational analysis of spectral functions, semidefinite programs, engineering problems, nonsmooth analysis and at least in quantum mechanics [3-7].

We recall that, matrices A,B∈Rn×n are similar, if B=S-1 AS with S∈ Rn×n invertible.

If B similar to A, then B and A have the same eigenvalues.

In the following, we reviewed some the formal definitions, theorems and examples of differentiable manifolds [8]. Also any book on the theory of differentiable manifolds may be used for reference if necessary.

Definition 1.1

1. A diffeomorphism f:Rn→Rn is an injection such that both f and its inverse f-1 are C function.

2. Let M,M′ are two differentiable manifolds, φ: M→ M′ be a mappi M ng, m∈M,φm∈M′. φ is called a submersion if its rank (rank of it’s Jacobian matrix) is equal to the dimension of M′ at each point of its domain. φ is called an immersion if its rank is equal to the dimension of M at each point of its domain.

3. A manifold M′ is said to be a submanifold of a manifold M if it is a subset of M and if the natural injection j:M′→ M is an immersion.

4. Let φ: M→ M′ be any function. A section of φ is a function ψ : M′→ M such that φ° ψ is the identity function on the domain of ψ.

For example, the set S(n,R) of real symmetric n×n matrices is a submanifold (with dimension n(n+1)/2) of the set of all real matrices M(n×n,R).

When the topology on M′ induced by its C structure is its topology as a subset of M, M′ is said to be a regular submanifold of M.

Next proposition is about submersions and it plays key role in the differentiability of eigenvalue function [8], propositions 6.1.2; 6.1.4; 6.2.1).

Proposition 1.1

1. φ is a submersion of M onto M′. If ψ: M′→ M″ is such that ψ° φ is differentiable then ψ also is differentiable.

2. A differentiable function φ: M→ M′ is a submersion if and only if for each m in its domain there exists a differentiable section of φ containing m in its range.

3. Let φ: M→ M′ is a submersion. If dim(M)>dim(M′), each set φ-1(s) has the structure of a regular submanifold of M of dimension dim(M)-dim(M′).

Definition 1.2

1. A Lie group G is a group which has the structure of a differentiable manifold and for which the group function

θ :G×G→G

defined by (g1,g2)→g1g2 is differentiable.

2. A Lie group is said to act on a differentiable manifold M as a Lie transformation group if we are given a global surjection

Φ :G×M→M

which is differentiable and such that if g,h∈G and m∈M

Φ(g,Φ(h,m))=Φ(gh,m).

If g∈G, the function Φg:M→M defined by m→Φ(g,m) is a diffeomorphism of M onto itself.

Let GL(n,R) be non-singular real matrices. The set O(n,R) of real orthogonal n×n matrices can be given the structure of a Lie subgroup of GL(n,R) of dimension image This Lie subgroup is a regular submanifold.

GL(n,R) acts on Rn as a Lie transformation group with the function Φ defined by (A,z)→Az.

Also, O(n,R) acts on Sn-1, unit sphere of Rn, as a Lie transformation group. This action is transitive.

Differentiability of λ(.)

In this section our aim is to prove that λ is a differentiable function between two differentiable manifolds. For this work, we will consider a C structure on the ordered non-increasing of elements of Rn. We start with a remark:

Remark 2.1

1. Let f:A→M be a bijection on a set A with values on a differentiable manifold M. Then f defines a complete C atlas of A. With this structure on A, f is a diffeomorphism [8].

2. Let n∈N (natural numbers)is a finite cardinal number and c be the cardinal of real number of R [9]. We recall that

(a) n+c=c

(b) c+c=c

(c) For each A,B sets

cardA+cardB=card(A∪B)+card(A∩B).

Where A, B are arbitrarily sets, cardA and cardB are their cardinal numbers.

Denote by Rn the ordered non-increasing of elements of Rn. That is,

image

It is clear that Rn is a convex, closed subset of Rn. The next result show that Rn is bijective with Rn.

Lemma 2.1: Let

image

Then Rn is bijective with Rn .

Proof 2: Let x=(x1,…,xn)∈Rn. There are n! (factorial)possible orders for x components. So there are n! permutation matrices as P1,…,Pn! gives rise to these orders in matrix notation. If P0 shows identity matrix and image then we have:

image

(Where P_i is a permutation,R^n is the set of n-tple of ordered real numbers, c is cardinal of real number set)

But image is bijective with Rn. So image By the above remark, part 2

image

image

But

image

So

image

image

Therefore image That is Rn bijective with Rn.

we have:

Corollary 2.3: Let

image

Then Rn admits a differentiable structure with dimension n.

Proof 2.4: Since Rn is a differentiable manifold, Corollary follows from the above Lemma and Remark 2.1, part 1.

We denote bijection in lemma 2.1 by image is a diffeomorphism and therefore if M is a submanifold of Rn, then T-1 (M) is a submanifold of Rn .

we define

image

(diag [] is a matrix with nonzero elements only in main diagonal.)

Thus diag[Rn ] is a differentiable manifold with dimension n.

Proposition 2.5: The eigenvalue function λ is a differentiable function.

Proof 2.6: The orthogonal Lie group O(n,R) acts as a Lie transformation group on S(n,R)as following

Φ :O(n,R)×S(n,R)→S(n,R)

(P,A)→PAPt

Where Pt is the transpose of matrix P.

So if P∈O(n,R), the function ΦP :S(n,R)→S(n,R) defined by A→Φ(P,A) is a diffeomorphism of S(n,R) onto itself and so it is a submersion. This follows that the function image defined by

(P,diag[s])→Pdiag[s]Pt

is a submersion. Let ρ be projection function on second factor

ρ:O(n,R)×Rn→S(n,R)

(P,diag[s])→diag[s],

then ρ is differentiable and ρ=λ° T . Hence, by the Proposition 1.1,part 1, λ is differentiable.

Proposition 2.5 has some important results about the properties of λ:

Corollary 2.7

1. λ is a submersion.

2. Let A∈S(n,R) with spectral decomposition A=Pdiag[λ1,…,λn] Pt. The Jacobian of λ, ∇λ, is given by

∇λAh=PhPt,

Where h∈S(n,R).

3. λ is an open mapping.

Proof 2.8

1. Let A∈S(n,R) and A=Pdiag[λ]Pt its spectral decomposition. If φ be Lie transformation group on Proposition 2.5, then

image

defined by φP(diag[s])=Pdiag[s]Pt is a differentiable section of λ. Hence, the state follows from Proposition 1.1, part 2.

2. It is well known that: If A is any point in the domain of a submersion λ :S(n,R)→Rn there exist charts x,y of S(n,R)→Rn at A,λ(A) respectively such that the function y° λ° x-1 is (z,w)→z.It is clear that y is identity function on Rn and x-1 is a P∈O(n,R) such that A=Pλ(A)Pt. We can write λ such that, on some neighborhood of A∈S(n,R) (P,diag[s])→diag[s]. In other words PAPt= diag[s]= λ(A). By multiplying both sides of this equality by Pt, P from the right and left, respectively, we have

Ptλ(A)P= A (1.1)

Suppose that A is fixed and h belong to a small neighborhood of A.With differentiating from both sides of (1.1) we have Pt∇λAP= I, where I is identity matrix in S(n,R). Therefore Pt∇λAP= h. This yields ∇λAP= PhPt.

3. Any submersion is an open mapping.

Example 2.1: The set M of all real symmetric matrices of order 2×2 with distinct eigenvalues is an open subset of S(2,R).So, by the above Corollary, λ(M) is an open subset of R2.

More Spectral Manifolds (Isotropic)

For fixed subset M ⊆ Rn, some properties on M remain true on the corresponding set λ-1(M)(spectral set). If M is differentiable manifold of Rn, then λ-1(M) will be called spectral manifold. The spectral manifolds are entirely defined by their eigenvalues [10].

For example, if the set M is symmetric, then properties such as closedness , convexity, prox-regular are transferred between M and λ-1(M) [11,12].

he set λ-1(M) often appears in engineering sciences, often as constraints in feasibility problems (for example, in the design of tight frames in image processing or in the design of low-rank controller in control) [13,14].

Also,transfer of differentiable structure of a submanifold M of Rn has been studied. We speculate that most of these results do’t depend on the property of symmetry. We must rewrite some of these results in the other work. In this section, we study fibers and orbits of λ as two types of spectral manifolds.

λ determines an equivalence relation ρ on S(n,R), defined by (A,B)∈ ρ if and only if λ(A)=λ(B). The equivalence classes of ρ are similar matrices. In fact, the following Lemma holds for any submersion .

Lemma 3.1: There is an equivalence relation ρ on S(n,R) such thatimage has a differentiable structure and it becomes a quotient manifold diffeomorphic to Rn .

This lemma has an important result about spectral manifolds. Let L be the diffeomorphism in the lemma and M⊆Rn be a submanifold. Then L-1(M) is a submanifold of image

That is:

Corollary 3.2: Any spectral submanifod in S(n, R) is union of similar matrices in S(n,R).

Similar matrices in S(n,R) have regular submanifold structure:

Lemma 3.3: For s Rn ∈ , λ-1(s) (fiber of λ)has the structure of a regular submanifold of S(n,R) of dimension image

Proof 3.4 Since λ is a submersion, Lemma follows from Proposition 1.4.

Let f given by Proposition 2.5. This Lie transformation group sets up an equivalence relation on S(n,R). The equivalence class containing a point A is the range of the function f A:O(n,R)S(n,R) (orbit of A) and

image

The set image is called stabilizer of A(or isotropy group). It is well known that if HA is not an open subgroup of O(n,R), then the orbit of A can be given the structure of a regular submanifold of S(n,R) diffeomorphic to the quotient manifoldimage Thus, by the following Lemma, λ-1(λ(A)) is a regular submanifold of S(n,R) of dimension

image

Lemma 3.5: For A∈S(n,R), the stabilizer of A is not an open subset of O(n,R).

Proof 3.6: Let HA be an open subgroup of O(n,R). Then the quotient set topology on image is discrete. Therefore λ-1(λ(A)) is discrete. This contradicts with Lemma 3.2.

Spectral Functions

Let A=Pdiag[λ1,…,λn]Pt, for some P∈O(n,R). In, Chen, Qi, and Tseng showed that for any function f :R→R, one can define a spectral function f:S(n,R)→S(n,R) [1,2,15] by

image (1.2)

which are constant on the orbit of A and the properties of continuity, directional differentiability, differentiability, and continuous differentiability are inherited by f from f. As we shall see, f does not play a large role.

In this section we examine only differentiability and continuous properties by using Lie group tools

Proposition 4.1: Let A∈S(n,R) and M be the orbit of A. Then differentiability of image not depends on the differentiability of f. not depends on the differentiability of f.

Proof 4.2: For each B∈M , λ(A)=λ(B). Let f° λi(B)=f(λi(B))=ci, i=,… ,n. If M′={B∈S(n,R)imageB=Pdiag[c1,…,cn]Pt} be the orbit of C=diag[c1,… ,cn] then the function f:M→M′ induced by f is defined

B→φ(P,C),∀P∈O(n,R)

where φ is Lie transformation action on the Proposition 2.5.Therefore f is differentiable.

Proposition 4.3: For function f :R→R, the following results hold:

1. f is differentiable at A∈S(n,R) with eigenvalues λ1,…, λn if and only if f is differentiable at λ1,…, λn.

2. . f is continuous at A∈S(n,R) with eigenvalues λ1,…, λn if and only if f is continuous at λ1,…, λn.

Proof 4.4

1. According to the second part of the Corollary 2.7, there exists an open neighborhood U of A∈S(n.R) and open neighborhood V of λ(A) in Rn such that λ( U)= V and λ in these local coordinates is the standard projection

λ(x11,…,x1n,…,xnn)=(x11,…,x1n).

Therefore we can write

f(A)=f(x11,…,x1n,…,xnn)=diag[f (x11),…,(x1n))]

That is f differentiable if and only if f differentiable.

2. This part is a consequence of part 1.

In the following proposition we have removed symmetric condition and other extra conditions [16,17].

Proposition 4.5: Suppose f:Rn→R be a function. Then the following assertions are true:

1. f is continuous at λ∈Rn if and only if f is continuous at Pλ for any P∈O(n,R).

2. f is differentiable at λ∈Rn if and only if f is differentiable Pλ for any P∈O(n,R).

Proof 4.6

1. O(n,R) acts on Rn as a transitively Lie transformation group(with matrix notation). Then ΦP:Rn→Rn by

ΦP(λ)=Pλ

is a diffeomorphism. But image Therefore two side of state are clear.

2. Proof of this part is similar to part (1).

In the sequel, corresponding with (1.1) , for each function f :Rn→Rn with

with
f (x)=(f1(x),…,fn(x))t (x∈Rn),

we define image by

image (1.3)

and similar to the Proposition 4.3 we can show that if f :Rn→Rn has the property of continuity(respectively, differentiability), then so does the symmetric-matrix function image

Note that image is independent of choice of P∈O(n,R) and that image(A) belongs to S(n,R), [18].

Proposition 4.7: Let f :Rn→Rn be a function with f (x)=(f1,…,fn(x)) t (a∈Rn), and let image be defined by (1.3). Let A∈S(n,R) with eigenvalue λ1(A),…,λn(A) and λ(A)=(λ1(A),…,λn(A))t . Then the following assertions are equivalent:

1. f is continuous at λ(A)

2. f is continuous at A.

Proposition 4.8: Let f :Rn→Rn be a function with f (x)=(f1,…,fn(x)) t (a∈Rn). Let A∈S(n,R) with eigenvalues λ(A)=(λ1(A),…,λn(A))t and λ(A)=(λ1(A),…,λn(A))t. Then the following assertions are equivalent:

1. f is differentiable at λ(A)

2. f is differentiable at A.

Acknowledgments

The research was supported by Marand Branch,Islamic Azad University.

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