The main theme of the thesis revolves around pseudospectra of matrix pencils and matrix polynomials and their applications in perturbation theory. Fourth, we present a general framework for the sensitivity analysis of eigenvalues of matrix pencils and matrix polynomials.
Introduction
Most importantly, we analyze the geometry of ill-conditioning eigenvalues of matrix pencils/polynomials. Finally, the sensitivity analysis of the eigenvalues of matrix pencils and matrix polynomials is presented in Chapter 5.
Preliminaries
Generalized eigenvalue problem
The set of n-by-n matrix pencils (homogeneous or inhomogeneous) is denoted by L(Cn×n). For the pencil L∈L(Cn×n) we say yes. Calculating the Schur decomposition of a matrix pencil is a well-posed problem and can be stably computed.
Polynomial eigenvalue problem
Then a mn-by-mn pencil X−zY is said to be a linearization of L if there exists a unimodular matrix polynomial E, F ∈ Lm(Cn×n) such that. Let A, B, C and D be matrices of suitable size such that matrix products AC and BD can be formed.
Subharmonic functions
Finally, we show that common boundary points of the components of pseudospectra of a matrix pencil are critical points. We show that a minimal critical point can be read from the pseudospectra of matrix pencils.
Preliminaries
We then show that if λ is a critical point of ηw,p, then there exists a pencil ∆L such that. We thus show that if λ is a critical point (generic or non-generic) of ηw,p, then λ is a multiple eigenvalue of the pencil lying on the boundary of the ηw,p(λ,L)-neighborhood of L.
A framework for pseudospectra of matrix pencils
For the rest of the chapter, we consider only the space Lpw(Cn×n,k · k) for appropriate choices ofw, p and the norm k · k onCn×n. Since the pseudospectrum of a pencil is determined by the geometry of the space to which the pencil belongs, for a regular pencil L ∈ Lpw(Cn×n,k · k), we can let denote the ²-pseudospectrum of L simply by Λ²(L) without having to specify w, pand k · k or showing the dependence of Λ²(L) on w, pand k · k.
Critical points and multiple eigenvalues
On the other hand, ifλ is a non-generic critical point of ηw,p(z,L) with multiplicity m, then λ is a multiple eigenvalue of L+ ∆L with geometric multiplicitym. The following result shows that the convergence points of components of Λ²(L) are actually critical points of ηw,p.
Wilkinson’s problem for matrix pencils
Note that if L ∈ Lpw(Cn×n,k · k2) is a regular pencil and has n distinct eigenvalues, then Theorem 2.4.11 provides a pencil that has a multiple value whenever a critical point of ηw,p(z,L) is available. A minimum critical point of ηw,p, that is, the critical point at which ηw,p takes the smallest value among all critical points, is of particular interest, and Theorem 2.4.13 tells us where to look for it. Further (λ, μ) is a multiple eigenvalue of L+ ∆L of geometric multiplicity 2 when σn is multiple and (λ, μ) is a degenerate eigenvalue of L+ ∆L whenσn is prime.
Diagonal pencils
Finally, we show that the common boundary points of the components of the pseudospectra of the matrix polynomial are the critical points. In particular, we show that the minimum critical point can be picked from the pseudospectra of matrix polynomials. Thus, we show that the solution of the Wilkinson problem for matrix polynomials can be read from the pseudospectra of the polynomials.
Preliminaries
Thus we show that if λ is a critical point (either generic or nongeneric) of ηw,p, then λ is a multiple value of a polynomial that lies on the limit of the ηw,p(λ,L)-neighborhood of L. In particular , when L is simple (i.e. has distinct eigenvalues) we show that a minimum critical point of ηw,p can be read from the pseudospectrum of L. Hence we show that the distance from L to the nearest polynomial that has a The multiple eigenvalue can be read from the pseudospectrum of L.
A framework for pseudospectra of matrix polynomials
Pseudospectra of matrix polynomials have been systematically developed and studied by Higham and Tisseur [25, 45]. However, the importance of this fact has not been emphasized enough in the literature when dealing with pseudospectra of matrix polynomials. As a consequence, pseudospectra of matrix polynomials as defined in the literature are ad hoc in nature.
Properties of pseudospectra of polynomial
This shows that the image of the component Λ²(revL) containing the eigenvalue 0 revL under the mapping z 7→z−1 is exactly the component Λ²(L) containing the infinite eigenvalue L. Then the limit of ∂Λ²(L ) of Λ²(L) is embedded in a real algebraic curve when p/(p−1) is an integer or p=∞. Consequently, we have the following result, which shows that the only local minimizers of ηw,p are the eigenvalues of L.
Critical points and multiple eigenvalues
The following result shows that critical points of ηw,p(z,L) are multiple eigenvalues of appropriately perturbed polynomials. On the other hand, if λ is a non-generic critical point of ηw,p(z,L) with multiplicity k, then λ is a multiple eigenvalue of L+ ∆L with geometric multiplicity k. This shows that common boundary points of components of pseudospectra of L are critical points of ηw,p(λ,L).
Wilkinson’s problem for matrix polynomials
Their analysis corresponds to the case p=∞ and their solution to Wilkinson's problem follows from Theorem 3.6.1 by setting p=∞. Recall that an infinite eigenvalue of L, if any, can be treated at the same level as finite eigenvalues considering the homogeneous form of L. We now present the analogue of Theorem 3.6.1 for homogeneous polynomials. Further (λ, µ) is a multiple eigenvalue of L+ ∆L of geometric multiplicity 2 when σ is multiple and (λ, µ) is a degenerate eigenvalue of L+ ∆L when σ is prime.
Introduction
We investigate inclusions of pseudospectra for matrix pencils and show that inclusions of pseudospectra can be usefully used to analyze the stability of self-decays. We present and analyze different notions of matrix pencil separation and show their usefulness in analyzing the stability of eigendecompositions. We investigate the conditions for the continuous evolution of eigendecompositions of L on a given open sphere with center in L. We show that a sufficient condition for the continuous evolution of eigendecompositions of L can be read from the pseudospectrum of L. We further show that for the corresponding norm the necessary and sufficient condition for the continuous evolution of the L eigendecays can be picked from the L pseudospectrum.
Localization of pseudospectra of matrix pencils
An eigendecomposition of Lis is said to evolve continuously on an open set U containing L ifX−1, Y and Lj are continuous on U. Continuous evolution of eigendecompositions of matrix pencils has been studied by Demmel et al. We investigate conditions for continuous development of self-decompositions of L on a given open ball centered at L. We show that a sufficient condition for continuous development of self-decompositions of L can be read from pseudospectra of L. Furthermore, we show that for a suitable norm a necessary and sufficient condition for continuous development of self-decompositions of L can be read from pseudospectra of L. Similar result holds for homogeneous matrix pencils. Cn×n, where I1 and I2 are the identity matrices of size k and n−k respectively, M ∈Ck×(n−k) and α, β are positive real numbers that are not zero.
Separation of pencils
Note that sep(L1,L2) as defined above is a natural generalization of the separation sep(A,B) of matrices. This shows that our definition of sep(L1,L2) not only generalizes the notion of separation sep(A, B) of matrices to the case of pencils, but it also unifies the notion of separation diff that exists for matrix pencils. The main advantage of our approach is that the separation of matrices and the separation of matrix pencils can be handled with equal ease.
Geometric separation and stable eigendecompositions
Obviously, (4.8) is ²-stable if and only if the eigenvalue from Λi and the eigenvalue from Λj do not move simultaneously and merge as L transforms into B(L, ²) for alli6=j. Proof: According to our construction, ηw,p(λ,L) is the smallest value of |||∆L|||w,p, so that λ is an eigenvalue of L+∆L. Assume that λ is not the common limit of the components of Λ²(L). Then |||∆L|||w,p = ηw,p(λ,L) and by Theorem 2.4.11, λ is a multiple eigenvalue of L+∆L. To prove the result, we show that the eigenvalue from λand the eigenvalues from Λ(L)\Λ simultaneously shift and merge at L→L+∆L.
Lower bounds of geometric separation
The above result shows that a lower bound of gsep provides a sufficient condition for²-stability of eigendecompositions of matrix pencils. We now derive the lower bounds of gsep according to the way we establish the relationship between gsep and sepλ.
Numerical examples
We show that our treatment unifies different measures of the sensitivity of simple eigenvalues of matrix pencils and matrix polynomials proposed in the literature. Letλ be a simple eigenvalue of a regular pencil L(z) = A−zB and y, x, respectively, be left and right eigenvectors of L corresponding toλ, i.e. L(λ)x= 0 and gy∗L(λ) = 0. How to measure the sensitivity of a simple eigenvalue λ (or (c, s)) to small perturbations in the pencil L.
Preliminaries
Our analysis not only provides a unified treatment of eigenvalue sensitivity analysis, but also makes the heart of the matter, so to speak, easy to understand and adds a new perspective to the sensitivity analysis of problems of polynomial eigenvalue. It is easy to see that the double rate of the p Schatten rate is the Schattenq rate. Since the spectral rate in Cn×n is the Schatten ∞ rate, we have the following result, the proof of which is immediate.
Condition Number
Obviously, the sensitivity of the solution S = f(a) to small perturbations in a in the direction of u, i.e. the sensitivity of S =f(a) in relation to the disturbance a+tu for small t is measured by kDuf(a) )kY. Then condu(f, a) := kDuf(a)kY is called the partial condition number of the solution S =f(a) with respect to u. Now the sensitivity of S =f(a) with respect to small perturbation ina in the direction of umaasured by condu(f, a). Therefore, the sensitivity of S =f(a) to small arbitrary perturbations in a is measured at sup.
Sensitivity of simple eigenvalues of a matrix
Recall that gsep(λ) is the smallest positive number for which the component of Λ²(A) that isolates λ from the rest of Λ(A) merges with another component of Λ²(A), when²→gsep(λ). Let nu² Recall that hX, Yi:= trace(Y∗X) defines an inner product on Cn×n. Also remember that L(Cn×n) is the vector space of pencils (without norm) of size n. Just in case one of the components of w is 0, the results of Theorem 5.5.4 still hold. We have shown that pseudospectra of matrix pencils/polynomials can be analyzed on the same lines as those of matrices. We have shown that a minimum critical point can be read from the pseudospectra of matrix pencils/polynomials. We have shown that our treatment unifies several measures of the sensitivity of simple eigenvalues of matrix pencils and matrix polynomials proposed in the literature.Sensitivity of simple eigenvalues of matrix pencils
Sensitivity of simple eigenvalues of matrix polynomials
