Some of these relate the Crawford number to the distribution of the eigenvalues of the definite pencil. In the case of the definite pencils, the distance to a nearest Hermitian pencil that is not definite is called the Crawford number [38, 29].
Notations
We also have estimates of the Crawford number of certain pencils in the form of upper and lower bounds. Some of these limits relate the Crawford number to the distribution of the pencil's eigenvalues.
Preliminaries
Regular matrix pencils
If S is a deflating subspace of L(z) of dimension r, then there exist non-singular matrices X and Y such that. Conversely, if (1.1) holds for some nonsingular matrices X and Y, then the first r columns of X span a deflating subspace of L(z).
Eigenvalue type and sign characteristic
A real eigenvalue of a Hermitian matrix polynomial is of definite type if it is either of positive type or negative type. The type of an infinite eigenvalue λ0 = ∞ of a Hermitian matrix polynomial P(z) is defined to be the type 0 as an eigenvalue of −revP(z).
Linearizations of matrix polynomials
The following result is very important in relating the eigenvalue types of real eigenvalues of Hermitian polynomials and their linearizations. Let λ be the eigenvalue of the Hermitian polynomial P(z) with the corresponding eigenvector x and let L(z) ∈ H(P) with the corresponding anzatz vector v be the linearization of P(z).
Homogeneous rotations
The following result from [4] relates the eigenvalue types of a Hermitian pencil to the corresponding eigenvalues of its rotated counterpart and shows that although rotations do not preserve the eigenvalue type, they always preserve the assigned type.
Backward error and ²-pseudospectrum
If any component C² of ΛH² (P) is a subset of the extended real line R∪ {∞}, it can be identified with a subset of the unit circle of the form. If P(z) is a regular Hermitian polynomial and C² is a component of ΛH² (P) such that C² ⊂R∪ {∞}, and ² is less than the distance to a nearest pencil which is not regular with respect to the norm not |k · |k, then the boundary ∂C² of C² is a finite subset of the extended real line.
Hermitian polynomials with real eigenvalues of definite type 19
They are defined as Hermitian quadratic matrix polynomials that satisfy one of the equivalent properties of Theorem 1.2.15. Proof: From the definition of ∂Γ and that of the backward error η(z, L) it follows that the curve Γ contains no eigenvalues of Lt(z) for all t ∈ ∂Γ.
Linear perturbation of definite pencils
The proof of the above theorem remains unchanged for the second case, even if both A and B are perturbed in such a way that the resulting pencil is final. Let Lt(z) be an n×n family of certain pencils, such that the union of the eigenvalues of Lt(z) forms an unconnected subset of the real line as t varies over an interval I on the real line.
Introduction
We propose a new definition of the eigenvalue type for a Hermitian polynomial P(z), based on the homogeneous form of the polynomial that determines the type of both finite and infinite eigenvalues in the same framework. This definition leads to another proof of the fact that homogeneous rotations do not create mixed-type eigenvalues. [4]) We also analyze certain properties of Hermitian pencils based on their canonical form under congruence, as given in (1.2). ² pseudospectrum of the pencil containing eigenvalues of perturbed pencils of only one type, positive or negative.
The Crawford number and distance problems
In particular, we show that the Crawford number is the distance to the nearest Hermitian pencil with a complex eigenvalue. We show that the Crawford number γ(A, B) is the distance from L(z) to the nearest Hermitian pencil with a mixed-type eigenvalue. If ∞ is an eigenvalue of L(z) of positive (negative) type, then 0 is an eigenvalue of −revL(z) of positive (negative) type.
If λ0 is a mixed-type semisimple eigenvalue, then L(z) is congruent to a block-diagonal pencil with a 2×2 block on the diagonal of the shape.
Homogeneous definition of eigenvalue type
Consider a homogeneous rotation of the Hermitian polynomial P(α, β) by an angle θ, which changes it to the polynomial ˜P(˜α,β).˜ Remember that (α, β) and (˜α,β) are related by. 3.5) wherec= cos θ and s= sin θ. The following lemma is crucial for relating the type of an eigenvalue to its rotated counterpart. This allows for unique representations of eigenvalues on the extended real line, with points both from the upper half and the lower half of the unit circle in R2. Note that the homogeneous definition of eigenvalue type limits the representations to the upper half of the unit circle for finite eigenvalues, while infinity corresponds to the point (1,0).
The rest of the proof follows after replacing ˜β0 by β0−sα0 in the above identity and dividing both sides by β0m−2.
Properties of Hermitian pencils
X1,m correspond to the blocks of the canonical form associated with real eigenvalues of the form. From the canonical form we see that the spans of the columns of X1,i and X1,j are A-orthogonal for i 6= j and that each of them individually is A-non-degenerate. If λ is complex, then there exists a Jordan chain corresponding to each Jordan block associated with λ, such that the vectors of the chain are mutually A-orthogonal and span an A-neutral subspace.
If λ is real or infinite and has a Jordan block of size m, then there exists a Jordan chain of length m such that the first bm2c vectors of the chain are A-orthogonal and an A-neutral span subspace.
Properties of Hermitian ²-pseudospectra
If λ = ∞, then it is of the positive (negative) type if for any matrix (negative) final. The following statements are equivalent. i) The component C² is of the positive (negative) type. ii) The eigenvalues of L(z) in C² are of positive (negative) type. If C² is a positive type component, then all eigenvalues of perturbed Hermitian pencils are (L+ ∆L)(z) with k|∆Lk| < ², in C² are of the positive type so that especially the eigenvalues of L(z) are also of the positive type.
The graph at the top in Figure 3.2 shows a component of ΛH² (L) of positive type containing the eigenvalue λ5 which merges at ∞ with another component of ΛH² (L) of positive type containing the eigenvalues λ1, λ2 and λ3 for ²= contains 1.7049.
Crawford number and Hermitian ²-pseudospectra
The finite eigenvalues of L(z) are of positive type if A is a positive definite matrix, of negative type if it is a negative definite matrix, and of positive and negative type if A is an indefinite matrix. In the latter case, the same holds for a Hermitian pencil closest to Lθ0(z). 3.18) In this case, if γ(A, B) < σmin(A), then it is the smallest value of ² for which a component of positive type iΛHγ(A,B)(L) joins a component of type negative. . Evidently, given Theorem 3.5.3, they are of positive type if A is positive definite and of negative type if A is negative definite.
Given a definable pencil L(z), dD(L) = min{²0, σmin(A)} where ²0 is the smallest value of ² for which the closure of ΛH² (L) has a component containing values eigenvalues of L(z) of positive and negative type.
Properties of Hermitian ²-pseudospectra
If all the eigenvalues of P(z) are real and of definite type, the degree of P(z) is at least 2 and ∞ is not an eigenvalue of P(z), then its finite eigenvalues are of both positive and negative type. Proof: All the eigenvalues of P(z) inC² are obviously of positive (negative) type if C² is of positive (negative) type. But this implies that all k eigenvalues of (L+t∆L)v(z) inC² are of positive type for all ∈[0,1]. This ensures that (P+∆P)(z) has k eigenvalues counting multiplicity in C².
Since ∆P(z) is an arbitrarily chosen Hermitian polynomial with k|∆Pk|< ², it follows that C² is a component of ΛH² (P) of positive type.
Distance problems via Hermitian ²-pseudospectra
Hyperbolic and quasihyperbolic polynomials
Suppose there exists a Hermite polynomial (P+ ∆P)(z) that has a complex eigenvalue, say z0, such that k|∆Pk|=dH(P). Thus, the Hermitian polynomial is closest to P(z) with respect to the norm k| · k|which is not hyperbolic, has either an infinite eigenvalue or a mixed-type real eigenvalue. Similarly, if P(z) is quasihyperbolic, then dQ(P) = σmin(Am) if it has eigenvalues of only positive or only negative type.
Therefore, there exists 0 < ² < σmin(Am) such that the termination of ΛH² (P) has at least one component containing eigenvalues of P(z) of both positive and negative types.
Overdamped quadratics
If P(z) has degree at least 2, then it has eigenvalues of positive and negative type by Proposition 4.2.1, and the rest of the proof follows by reasoning as in the case of hyperbolic polynomials. ²∈(0, λmin(A2)), no component of the closure Λ²(Q) contains ∞. Also, all eigenvalues of Q(z) in any component are of positive or negative type. The definition²0 implies that the closure ΛH²0(Q) has a component, say C²0, which contains eigenvalues Q(z) of positive and negative type and is formed from the union of components ΛH² (Q) such that the eigenvalues Q(z ) belonging to them are of positive or negative type.
Therefore, by Theorem 4.2.4, C²0 is formed by the fusion of components of Λ²0(Q) of positive and negative type and by Theorem 4.2.8, any fusion point of these components is an eigenvalue of mixed type of a Hermitian polynomial ( Q+∆Q)(z) true.
Definite polynomials
Let P(z) be a certain polynomial of degree m ≥ 2 such that one of the following conditions holds. Furthermore, in view of Theorem 4.2.4, the sum of the multiplicity of the eigenvalues of (P + ∆P)(z) in C² is constant for . If each of the components of the closure of ΛH² (P) contains only eigenvalues from one of these groups and.
To prove the equality, let ˆC²0 be the component of the closure ΛH²0(P), which contains an eigenvalue from one of the groups Ej and Ej−1 for j ∈ {1,2,.
Computing the Crawford number and other distances
Note that in most cases we observe a very good lower bound on the Crawford number within a few iterations. The second column of the table gives the point z0 represented by (cos θ0,sinθ0) where θ0 is the point at which the maximum of ismin(Aθ) occurs (giving the Crawford number). Table 5.2 gives the Crawford number for different pencil sizes given A−zB where A = |i−j| (Fiedler matrix).
For sizes 640 and 1280, the outputs were produced with tol1 =|kL|k ×10−14. The second column gives the point z0 represented by (cos θ0,sin θ0), where θ0 is the point of I where the maximum of σmin (Aθ) (which gives the Crawford number) is achieved.
Bounds for the Crawford number
If this is written in terms of the homogeneous form of the backward error, we have. Then δ is a measure of the separation between the sets of positive and negative type eigenvalues with respect to the chord metric. We have performed a Hermitian ²-pseudospectra based analysis of the distance of certain classes of Hermitian matrix polynomials with real eigenvalues of a given type to a nearest Hermitian polynomial outside the class with respect to a pre-specified norm.
Some of these limits relate the Crawford number to the distribution of the eigenvalues of the particular pencil.
Eigenvalue and singular value curves for Example 3.5.12
Eigenvalue and singular value curves for Example 3.5.13
Eigenvalue and singular value curves for Example 3.5.14
Plots of η(θ, Q) and g(θ) for Example 4.2.10
Plots of η(θ, Q) and g(θ) for Example .11
Plots of η(θ, Q) and g(θ) for Example 4.2.12
