Lie Group Methods for Eigenvalue Function

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.


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 1  (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][4][5][6][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. 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).  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.

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 R n . We start with a remark: 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 Where A, B are arbitrarily sets, cardA and cardB are their cardinal numbers.
Denote by n R ≥ the ordered non-increasing of elements of R n . That is, It is clear that n R ≥ is a convex, closed subset of R n . The next result show that n R ≥ is bijective with R n .

Lemma 2.1: Let
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

More Spectral Manifolds (Isotropic)
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.

Spectral Functions
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 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 where φ is Lie transformation action on the Proposition 2.5. Therefore f ◊ is differentiable. Therefore we can write f ◊ (A)=f ◊ (x 11 ,…,x 1n ,…,x nn )=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].