Rafikul Alam, Professor, Department of Mathematics, Indian Institute of Technology Guwahati for the award of the degree of Doctor of Philosophy and this work has not been submitted elsewhere for the degree. Saurav Choudhury of the department for their help in various ways during my research period. For a general norm on MEP space, we define the conditional number cond(λ,W) of the simple eigenvalue λ MEPW and derive three equivalent representations of cond(λ,W). We also analyze the holomorphic perturbation of a simple eigenvalue W. We further define the backward error η(λ,W) of the approximate eigenvalueλofW.
We define and analyze two vector spaces, namely the right ansatz space and the left ansatz space of potential linearizations of a two-parameter matrix polynomial of the form P(λ, µ) :=Pk.
Preliminaries
Vector space of linearizations of matrix polynomial
Then x ∈ Cn is a straight eigenvector of P corresponding to the finite eigenvector λ ∈ C if and only if Λ ⊗x is an eigenvector of L corresponding to the eigenvalue λ. In fact, every right eigenvector of L with finite eigenvector λ is of the form Λ⊗x for some eigenvector of P corresponding to λ. Similarly, if L ∈ L2(P) with nonzero left ansatz vector w, then y ∈ Cn is a left vector of P corresponding to the finite eigenvalue λ ∈ C if and only if Λ⊗y is a left vector of L corresponding to its eigenvalue. λ.
The right and left ansatz spaces provide a broad class of matrix pencils corresponding to a given matrix polynomial.
Nonsingular MEP
In [42] it is shown that all matrix pencils in the right and left ansatz space are not linearizations of the given matrix polynomial. The commuting tuple of operators S= (S1, · · · , Sm) is said to be the associated system of the homogeneous MEPW. Atkinson[12] proved that the spectrum of a non-singular homogeneous MP is the same as the spectrum of the MP's corresponding associated system.
The moving tupleT = (T1,· · · , Tm) is said to be the companion system of the non-singular non-homogeneous MEP W .
Nonhomogeneous MEP
Sensitivity analysis of simple eigenvalues
Then the eigenvalue λW and an eigenvalue µW 6= λW of the MEP W must move and merge at λ(X) whenb X changes from W to X. When MEP(n1,· · · , nm) is fitted with the weighted norm with non-zero positive weights, we have the following result.
Holomorphic perturbations
Then there is an open neighborhood nbd(bt) ⊂ Cp containing bt and a holomorphic function λ : nbd(bt) → Cm such that λ(bt) = bλ and λ(t) is a simple eigenvalue of W(t, z ) for all t ∈ nbd(bt). Therefore, according to implicit function theorem, there exists an open neighborhood nbd(bt)⊂Cp containing bt and a holomorphic function λ: nbd(bt)→ Cm, such that λ(bt) =λb and φ(t, λ(t)) = 0 , for all t∈ nbd(bt). From the similar arguments to those in the proof of Theorem 2.3.1, it follows that λ(t) is a simple eigenvalue of W(t, z), for all t∈nbd(bt).
Backward perturbation analysis
In this chapter, we undertake sensitivity and posterior perturbation analysis of a two-parameter polynomial eigenvalue problem of degree k.
Polynomial MEP
For perturbation analysis of a regular two-parameter polynomial eigenvalue problem of degreek, we define norms, inner product and double norms onPMEP(k;n1, n2) as follows.
Sensitivity analysis of a simple eigenvalue
The regular two-parameter inhomogeneous polynomial MEP of degree k reduces to a regular matrix polynomial W(z) := Pk. Furthermore, if x := x1 ⊗x2 and y := y1 ⊗y2 are right and left eigenvectors of W corresponding to λW, respectively, then we have. When PMEP(k;n1, n2) is equipped with the weighted norm |||·||w,V with non-zero positive weights, we have the following result.
Given a weighted norm|||·|||w,V onPMEP(k;n1, n2), we construct a fast perturbation for a simple eigenvalue λW ∈σ(W). Fast perturbation) Let W∈PMEP(k;n1, n2) be regular and λW ∈ σ(W) be simple.
Holomorphic perturbations
Then, from arguments similar to those given in the proof of Theorem 2.4.1, it follows that there is an open neighborhood nbd(bt)⊂Cp containing bt and a holomorphic functionλ: nbd(bt)→Cm such that λ(bt) = bλ and λ(t) is a simple eigenvalue of W(t, z) for all t ∈ nbd(bt).
Backward perturbation analysis
In this chapter, we develop a framework for sensitivity and inverse perturbation analysis of a regular homogeneous MEP W. More precisely, we define the conditional number cond(λ,W) of a simple eigenvalue λ for W and derive various representations of cond(λ ,W). In particular, if λ∈σ(W) is geometrically simple, then there are left and right eigenvectors ub:=bu1⊗ · · · ⊗bum and bv :=bv1⊗ · · · ⊗bvm of W corresponding to λ, that's it.
Note that by the simple zero of Gλ(z) we mean the solution Gλ(z) = 0 where the derivative DGλ(z) is non-singular. Then the following conditions are equivalent:. iv) NW(λ;y, x) is nonsingular for all left and right eigenvectors y, x of W corresponding to λ. By Proposition 4.2.4, there exist left and right eigenvectors bu and bv of W corresponding to λ such that NW(λ) = NW(λ;u,b bv).
This contradicts the assumption that NW(λ;y, x) is non-singular for all left and right eigenvectors y and x of W corresponding to λ respectively. Again, according to Proposition 4.2.4, there exist left and right eigenvectors bu and bv of W corresponding respectively to λ, so that NW(λ) = NW(λ;bu,bv). In this section, we perform a sensitivity analysis of an algebraically simple eigenvalue of a regular homogeneous MEP and derive the condition number from a simple eigenvalue.
Equivalently, ifx:=x1⊗· · ·⊗xm andy:=y1⊗· · ·⊗ym are right and left eigenvectors of W corresponding to λW respectively, then it holds. Now we construct a fast perturbation for a simple eigenvalue of a homogeneous MEP, considering a weighted norm for MEP(n1, · · · , nm).
Holomorphic perturbations
Then there is an open neighborhood nbd(bt) ⊂ Cp containing bt and a holomorphic function λ : nbd(bt) → Cm+1 such that λ(bt) =bλ and λ(t) is a simple eigenvalue of W(t, z) for all t ∈nbd(bt). Therefore, by implicit function theorem, there is an open neighborhood nbd(bt) containing bt and a holomorphic function λ: nbd(bt)→Cm+1, such that λ(bt) = bλ and φ(t, λ(t)) = 0 for all t ∈nbd(bt).
Backward perturbation analysis
Linearization is a process that transforms a polynomial eigenvalue problem into a generalized eigenvalue problem of larger size. In this chapter, our main goal is to construct linearizations of the two-parameter polynomial eigenvalue problem. We derive the linearization conditions for an element in the right or left anzatz space.
Ansatz spaces for two-parameter polynomials
Right and left ansatz spaces
This motivates us to focus on two-parameter matrix pencils L(λ, µ) such that L(λ, µ)·(Λ⊗In) =v⊗P(λ, µ) for some v ∈Cp. In the remainder of this chapter, we follow the convention that A(i, j) is the (i, j)th block of Whenever A is a matrix of blocks, each block being an ann×n matrix. Sum shifted into column) Let X, Y, Z be p×p block matrices with each. The column-shifted sum of three p×p block matrices is a p×q block matrix, which is characterized in the following result.
Next, we derive the column-shifted sum of the coefficient matrices of the first standard form L1(λ, µ). For a quadratic two-parameter matrix polynomial we have the following result which is proved in [1]. Now using Lemma 5.2.2, we characterize a linear two-parameter matrix polynomial L(λ, µ)∈L1(P) with right ansatz vector v ∈ Cp.
To characterize the left anzatz space, we define a row-shifted sum as follows. row-shifted sum) Let X, Y, Z be p × p block matrices with n × n matrices as blocks. The row-shifted sum of three p × p block matrices is a q × p block matrix, which is characterized in the following result. We then derive the row-shifted sum of the coefficient matrices of the second standard form given in (5.14).
Next, we characterize a two-parameter matrix pencilL(λ, µ)∈L2(P) with left ansatz vector w via row shift the sum of the coefficient matrices of L(λ, µ). Let L(λ, µ) = λX +µY +Z be a two-parameter matrix pencil such that X, Y, Z are 3×3 block matrices, where each block is an n×n matrix.
Procedure for determining the linearization condition for a two-parameter matrix pencil in L1(P):. 3) Apply the appropriate block transformation M ⊗In to L(λ, µ).
Double ansatz space
Linearizations of a two-parameter PEP
Since structured MEPs appear in many applications, in this chapter we undertake the analysis of structured feedback disturbances of structured MEPs. We consider twelve special structured MEPs, namely T-symmetric, T-skewed symmetric, T-even, T-odd, T-even alternating, T-odd alternating, H-Hermitian, H-oblique Hermitian, H-even, H- odd, H-even alternating and H-odd alternating.
Structured MEPs
Then (λ, x, y) is an eigentriple of W if λ is an eigenvalue of W and x and are right and left eigenvectors of W corresponding to λ respectively, that is, W(λ)x= 0 and yHW(λ) = 0. Note. that in this case xTkBkjxk= 0 when j is odd and therefore it follows thatxTkrk =−xTkAkxk− P. We prove the following result with respect to the smallest p-norm solutions of a system of linear equations that we will use in the sequel.
Frobenious norm structured backward error
Since Qk is unitary and QHkxk = e1, the first column of the identity matrix, we have. Thus it follows that the maps H-Hermitian → H-skew-Hermitian given by W 7→ iW and H-Skew-Hermitian → H-Hermitian given by X 7→ iX are isometric isomorphisms. Combining these facts, it turns out that the structured backward error of (λ, x) as an approximate eigenpair of a skew-Hermitian MEP can be obtained from the structured backward error of (λ, x) as a pair approximate a Hermitian MEP and vice versa.
By similar arguments, we can further show that the backward structured error of (λ, x) as an approximate pair of an H-odd MEP (respectively H-odd alternation) can be obtained from the backward structured error of ( λ, x ) as an approximate eigenpair of a MEP H-pair (respectively alternating H-pair) and vice versa. We now derive the structured backward error of (λ, x) as an approximate pair of H-Hermitian MPs, H-even and alternating H-even considering the oldest H¨norm 2 |||·||| 2 over MEP(n1, · · · , nm).
Spectral norm structured backward error
In a similar way, in the case of aH-even alternating MEP, we have the following result. For the general norm on the MEP space, we defined the conditional number cond(λ,W) of the simple eigenvalue λ MEP Wby. Given a simple eigenvalue λW of W, we have shown that there exists an open neighborhood nbd(W) containing W and a smooth function λ such that λ(W) = λW and λ(X) is a simple eigenvalue of X for all X ∈nbd(W).
We determined the derivative Dλ(W) and used it to derive three equivalent representations of cond(λ,W). We also constructed a fast perturbation for λW taking into account the weighted norm in the MP space. We defined and analyzed two vector spaces of potential linearizations of a two-parameter polynomial of degree k.
Finally, we have considered structured linear MEPs and analyzed structured backward errors and structured backward perturbations. We have denoted the space of structured MEPs of True defined structured backward error of an approximate eigenpair (λ, x:=x1 ⊗ · · · ⊗xm) by. Alam, On backward errors of structured polynomial eigenproblems solved by structure-preserving linearizations, Linear Algebra Appl p.1989-2017.
Plestenjak, Backward error, condition number and pseudospectra for the multiparameter eigenvalue problem, Application of Linear Algebra pp.63-81. Ko˘sir, Kronecker bases for linear matrix equations, with application to two-parameter eigenvalue problems, Application of Linear Algebra pp.259-288.
