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We present in this paper a new method for solving polynomial eigenvalue problem. We give methods that decompose a skew-Hamiltonian matrix using Cholesky like-decomposition. We transform first the polynomial eigenvalue problem to an equivalent skew-Hamiltonian/Hamiltonian pencil. This process is known as linearization. Decomposition of the skew-Hamiltonian matrix is the fundamental step to convert a structured polynomial eigenvalue problem into a standard Hamiltonian eigenproblem. Numerical examples are given.

In this work we propose a numerical approach for solving the k^{th} degree polynomial eigenvalue problem

P ( λ ) v = ∑ i = 0 k λ i M i v = 0 (P)

Problem (P) arises in many applications in science and engineering, ranging from the dynamical analysis of structural systems such as bridges and buildings to theories of elementary particles in atomic physics [^{th} degree polynomial eigenvalue problem (P) to an equivalent first-degree equation ( A − λ B ) v = 0 commonly called pencil problem. In the case when matrices M i have symmetric/skew-symmetric structure, the problem (P) is transformed to a skew-Hamiltonian/Hamiltonian pencil [

The Hamiltonian matrix H is given by J T R − T A R − 1 where J = ( 0 I n − I n 0 ) .

It is known that any nonsingular skew-symmetric matrix has a decomposition of the form B = R T J R [

We give in this paragraph, new definitions and useful propositions.

Let J = J 2 n = ( 0 I n − I n 0 ) , where I n denotes the n × n identity matrix. We

will use J when the size is clear from the context. Recall that a matrix M ∈ ℝ 2 n × 2 n is skew-Hamiltonian if M J = M , where the J-transpose of the matrix M is defined

by M J = J T M T J . Likewise, a Hamiltonian matrix H is written as ( E G F − E T )

where E, G and T ∈ ℝ n × n with G T = G and F T = F . We have H J = − H . More general, the J-transpose of the rectangular 2p-by-2q matrix N is defined by 2q-by-2p matrix N J = J 2 q T N T J 2 p .

The set ( E i ) 1 ≤ i ≤ n where E i = [ e i e n + i ] with e i is denoting the i-th unit vector of length 2n, satisfies E i J 2 = J 2 n E i , E i J = E i T and E i T E j = δ i j I 2 where E i J = J 2 T E i T J 2 n and δ i j = { 1 if i = j 0 if i ≠ j

Let U = [ u 1 , u 2 ] ∈ R 2 n × 2 where u 1 = ∑ i = 1 2 n u i 1 e i and u 2 = ∑ j = 1 2 n u j 2 e j . Then U is written in a unique way as linear combination of ( E i ) 1 ≤ i ≤ n on ℝ 2 × 2 , U = ∑ i = 1 n E i M i where M i = ( u i 1 u i 2 u n + i 1 u n + i 2 ) . Let M ∈ ℝ 2 n × 2 n be a 2n-by-2n real matrix. Then M is written as M = ∑ i = 1 n ∑ j = 1 n E i M i j E j T where M i j = ( m i j m i , n + j m n + i , j m n + i , n + j ) .

Definition 2.1. The 2n-by-2n real matrix L = ∑ i = 1 n ∑ j = 1 n E i L i j E j T is called lower J-triangular if L i j = 0 2 × 2 for j > i and L i i = ( ∗ 0 ∗ ∗ ) , (i.e., L = ∑ i = 1 n ∑ j = 1 i E i L i j E j T ).

Definition 2.2. The 2n-by-2n real matrix U = ∑ i = 1 n ∑ j = 1 n E i U i j E j T is called upper J-triangular if U i j = 0 2 × 2 for i > j and U i i = ( ∗ ∗ 0 ∗ ) , (i.e., U = ∑ i = 1 n ∑ j = i n E i U i j E j T ).

Proposition 2.1. Let M and N be two upper J-triangular (respectively, lower J-triangular) 2n-by-2n real matrix. The product

Proof. Let

That proves

Definition 2.3.

Proposition 2.2. The inverse of a regular upper J-triangular 2n-by-2n real matrix (respectively, lower J-triangular) is also upper J-triangular (respectively, also lower J-triangular).

Proof. Let

Since

with

In this section, we study different ways to compute

Definition 3.1. The 2n-by-2n real skew-Hamiltonian matrix M is called J-definite if

Remark 3.1 For

Lemma 3.1. If M is a 2n-by-2n real skew-Hamiltonian and J-definite matrix, then M is invertible.

Proof. If not, there exists

Theorem 3.2. If M is a 2n-by-2n real skew-Hamiltonian, J-definite matrix, then M has an LU J-factorization.

Proof. Let

Corollary 3.3. If

Proof. Since the matrix M is skew-Hamiltonian, then by taking

Theorem 3.4. Let M be a 2n-by-2n real skew-Hamiltonian J-definite matrix, then M has a Cholesky J-factorization

and in addition the

Proof. We proceed by induction on n. For

Let’s now

We set

Since

Since

We construct an algorithm that gives decomposition

Proposition 3.5. Let M is a 2n-by-2n real skew-Hamiltonian, J-definite matrix. If

(here

Proof. By corollary 3.3,

where

We study now a method that constructs decomposition

Let

with

Since

then

If

And for

Thus

Since

then,

Since

If we set

However

The method yield the following algorithm.

Algorithm:

for

for

Many applications give rise to structured matrix polynomial eigenvalue problems

The numerical solution of this polynomial eigenvalue problem is one of the most important tasks in the vibration analysis of buildings, machines and vehicles [

Theorem 4.1. [

and

We draw from this theorem that the polynomial eigenvalue problem (P) can be reduced to an eigenvalue pencil problem

We present computed eigenvalues that solve the k^{th} degree polynomial eigenvalue problem

Example 1. [

We obtain a 144 × 144 quartic pencil, whose 576 eigenvalues are shown in

Example 2. [

The 400 eigenvalues are shown in

We have proposed a numerical approach for solving polynomial eigenvalue problems structured. We first transform polynomial eigenvalue problem

We thank the editor and the referee for their comments.

The author declares no conflicts of interest regarding the publication of this paper.

Bassour, M. (2020) Hamiltonian Polynomial Eigenvalue Problems. Journal of Applied Mathematics and Physics, 8, 609-619. https://doi.org/10.4236/jamp.2020.84047