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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.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

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

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 are real, and it admits a spectral decomposition of the form

(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[λ_{1},…,λ_{n}] is the diagonal matrix
with its ith diagonal entry λ_{i} and P^{t} 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)→R^{n} be the eigenvalue function such that λ_{i}(A),i=1,…
,n, yield eigenvalues of A for any A∈S(n,R) and are ordered in a
nonincreasing order, that is, λ_{1}(A)≥…≥λ_{n}(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∈R^{n×n} are similar, if B=S^{-1} AS with S∈
R^{n×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:R^{n}→R^{n} 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 (g_{1},g_{2})→g_{1}g_{2} 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 Φ_{g}: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 This Lie subgroup is a regular submanifold.

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

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

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 R^{n}. 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 R^{n}_{≥} the ordered non-increasing of elements of R^{n}. That is,

It is clear that R^{n}_{≥} is a convex, closed subset of R^{n}. The next result
show that R^{n}_{≥} is bijective with R^{n}.

**Lemma 2.1: Let**

Then R^{n} is bijective with R^{n}_{≥} .

**Proof 2:** Let x=(x_{1},…,x_{n})∈R^{n}. There are n! (factorial)possible orders
for x components. So there are n! permutation matrices as P_{1},…,P_{n}!
gives rise to these orders in matrix notation. If P_{0} shows identity matrix
and then we have:

(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 is bijective with R^{n}_{≥}. So By the
above remark, part 2

But

So

Therefore That is R^{n}_{≥} bijective with R^{n}.

we have:

**Corollary 2.3:** Let

Then R^{n}_{≥} admits a differentiable structure with dimension n.

**Proof 2.4:** Since R^{n} is a differentiable manifold, Corollary follows
from the above Lemma and Remark 2.1, part 1.

We denote bijection in lemma 2.1 by is a
diffeomorphism and therefore if M is a submanifold of R^{n}, then T^{-1} (M) is a submanifold of R^{n} .

we define

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

Thus diag[R^{n}_{≥} ] 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)→PAP^{t}

Where P^{t} 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 defined by

(P,diag[s])→Pdiag[s]P^{t}

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

ρ:O(n,R)×R^{n}→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[λ_{1},…,λ_{n}]
P^{t}. The Jacobian of λ, ∇λ, is given by

∇λAh=PhP^{t},

Where h∈S(n,R).

3. λ is an open mapping.

**Proof 2.8**

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

defined by φP(diag[s])=Pdiag[s]P^{t} 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)→R^{n} there exist charts x,y of S(n,R)→R^{n} 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 R^{n} and x^{-1} is a
P∈O(n,R) such that A=Pλ(A)P^{t}. We can write λ such that, on
some neighborhood of A∈S(n,R) (P,diag[s])→diag[s]. In other
words PAP^{t}= diag[s]= λ(A). By multiplying both sides of this
equality by P^{t}, P from the right and left, respectively, we have

P^{t}λ(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 P^{t}∇λAP= I,
where I is identity matrix in S(n,R). Therefore P^{t}∇λAP= h. This yields
∇λAP= PhP^{t}.

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 R^{2}_{≥}.

For fixed subset M ⊆ R^{n}, some properties on M remain true on the
corresponding set λ^{-1}(M)(spectral set). If M is differentiable manifold
of R^{n}, 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 R^{n} 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 that has a differentiable structure and it becomes a quotient manifold
diffeomorphic to R^{n} .

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

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 R^{n}_{≥} ∈ , λ^{-1}(s) (fiber of λ)has the structure of a
regular submanifold of S(n,R) of dimension

**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

The set 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 manifold Thus, by the following
Lemma, λ^{-1}(λ(A)) is a regular submanifold of S(n,R) of dimension

**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 is discrete. Therefore λ^{-1}(λ(A)) is discrete. This
contradicts with Lemma 3.2.

Let A=Pdiag[λ_{1},…,λ_{n}]P^{t}, 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

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 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° λ_{i}(B)=f(λ_{i}(B))=c_{i}, i=,…
,n. If M′={B∈S(n,R)B=Pdiag[c_{1},…,c_{n}]P^{t}} be the orbit of C=diag[c_{1},…
,c_{n}] 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 λ_{1},…, λ_{n} if and
only if f is differentiable at λ_{1},…, λ_{n}.

2. . f^{◊} is continuous at A∈S(n,R) with eigenvalues λ_{1},…, λ_{n} if and
only if f is continuous at λ_{1},…, λ_{n}.

**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 R^{n}_{≥} such that λ( U)= V and λ in these local
coordinates is the standard projection

λ(x_{11},…,x_{1}n,…,x_{n}n)=(x_{11},…,x_{1}n).

Therefore we can write

f^{◊}(A)=f^{◊}(x_{11},…,x_{1}n,…,x_{n}n)=diag[f (x_{11}),…,(x_{1n}))]

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:R^{n}→R be a function. Then the following
assertions are true:

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

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

**Proof 4.6**

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

ΦP(λ)=Pλ

is a diffeomorphism. But 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 :R^{n}→R^{n} with

with

f (x)=(f_{1}(x),…,f_{n}(x))t (x∈R^{n}),

we define by

(1.3)and similar to the Proposition 4.3 we can show that if f :R^{n}→R^{n} has
the property of continuity(respectively, differentiability), then so does
the symmetric-matrix function

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

**Proposition 4.7: **Let f :R^{n}→R^{n} be a function with f (x)=(f_{1},…,f_{n}(x))
t (a∈R^{n}), and let be defined by (1.3). Let A∈S(n,R) with eigenvalue
λ_{1}(A),…,λ_{n}(A) and λ(A)=(λ_{1}(A),…,λ_{n}(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 :R^{n}→R^{n} be a function with f (x)=(f_{1},…,f_{n}(x))
t (a∈R^{n}). Let A∈S(n,R) with eigenvalues λ(A)=(λ_{1}(A),…,λ_{n}(A))t and
λ(A)=(λ_{1}(A),…,λ_{n}(A))t. Then the following assertions are equivalent:

1. f is differentiable at λ(A)

2. f is differentiable at A.

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

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