At present an Iranian young athlete Ali Darvish is one of the best players of ..... Iranians, Americans, Canadians, Spanish, Chinese, Japanese, etc, mathematians ...
August 27th(9:00-10:00) 2008; Kerman, Iran
the generalized numerical range of matrix polynomials
Hiroshi Nakazato (Hirosaki University, JAPAN)
Abstract In this talk, the generalized numerical range of matrix polynomial is introduced and its properties are discussed. Especially the algorithm to compute the boundary of the generalized numerical range is provided in some cases. Contents 00. Introduction 0. Motivations or Aims 1. Numerical objects of matrices and matrix polynomials 2. Numerical range of a matrix as V 1 (A) 3. When I was younger,... 4. Weighted shift operators 5. (1) C-numerical range (2) application to NMR 6. Kippenhahn’s method to compute the boundary 7. (1) Mathematics and Philosophy (2) Al-Tusi (3) Epicycles and the k-numerical range 1
8. Joint numerical range and its convex hull 9. Polynomial numerical hulls 10. Conclusion
Fig.1 Lion stone carving at Persepolis
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00. Introduction I would like to express my thanks to the organizers of this conference for inviting me and giving me an oppotunity to talk here. At first I call that Iran and Japan have a long history of their frienship. About 1600 years ago, the first Dynasty or State in Japan was built at Nara. The photo in the next page is our national treasure which was brought from the Sasanian Dynasty via China about 1250 years ago.
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Fig.2 a National Trasure of Japan
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That is an excellent cut glass. At present an Iranian young athlete Ali Darvish is one of the best players of Japanese professional sports.
Fig.3 Ali Darvish
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I hope that this visit promote our study on Mathematics and increase our friendship.
Fig.4 a map of Asia
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Fig.5 Mt.Fuji in Japan
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0. Motivations or Aims Matrix Analysis has many applications. 1: the application of the eigenvalue problem to the analysis of the oscillations of buildings. 2: the applications to the analysis of simultaneous differential equations. 3: Localization of the eigenvalues of a linear operator
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In Iran and Japan, earth-quakes frequently occur. We are intersted in the structural performace of building against earth-quakes.
Fig.6 building destroyed by an earth-quake
In [19;2005], my colleague Tsumura and I discussed the strength of concrete columns against horizontal loads using numerical ranges. I hope that our study in Matrix Analysis contributes to the safety and economic activities of nations. 9
1. Numerical objects of matrices and matrix polynomials Let Mn be the associative algebra of all n × n complex matrices. Supose that Q(λ) = Am λm + Am−1 λm−1 + · · · + A1 λ + A0 is a matrix polynomial, where Ai ∈ Mn (i = 0, 1, . . . , m) and λ is a complex variable. Many numerical objects are defined for a matrix A ∈ Mn or a matrix polynomial Q(λ). Probably, the most important one is their eigenvalues or the spectra. The spectrum σ(A) of a matrix A is defined as σ(A) = {λ ∈ C : Aξ = λξ for some ξ ∈ Cn , ξ 6= 0} = {λ ∈ C : det(λIn − A) = 0}. 10
(1.1)
The spectra of matrices have good properties such as σ(p(A)) = p(σ(A)) for a polynomial p(λ) and σ(AB) = σ(BA). Let A = U |A| be the polar decomposition of A.The Schatten p-norm ||A||p of a matrix A is defined as
Fig.7 p=4
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||A||p = tr(|A|p )1/p
(1.2)
( 1 ≤ p < ∞). We also consider the operator norm ||A|| defined as p ||A|| = max{ (Aξ)∗ (Aξ) : ξ ∈ Cn , ξ ∗ ξ = 1} √ = max{ λ : λ ≥ 0, det(λIn − A∗ A) = 0}.
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(1.3)
We consider the case A = diag(a1 , a2 ) with a1 , a2 are real numbers. We consider the convex set Ω = {(a1 , a2 ) ∈ R2 : ||diag(a1 , a2 )||p ≤ 1} = {(a1 , a2 ) ∈ R2 : |a1 |p + |a2 |p ≤ 1} and its boundary ∂Ω in R2 . The curve ∂Ω is an algebraic curve, i.e. , there is a real polynomial f (x, y) ∈ R[x, y] s.t. f (a1 , a2 ) = 0 for |a1 |p + |a2 |p = 1 ⇔ p is a rational number. The spectrum σ(Q(λ)) of a matrix polynomial Q(λ) is defined as σ(Q(λ)) = {λ ∈ C : Q(λ)ξ = 0, for some ξ ∈ Cn , ξ 6= 0} = {λ ∈ C : det(Q(λ)) = 0} = {λ ∈ C : 0 ∈ σ(Q(λ))}. 13
(1.4)
2. Numerical range of a matrix as V 1 (A) The numerical range W (A) of a matrix A ∈ Mn is defined as W (A) = {ξ ∗ Aξ : ξ ∈ Cn , ξ ∗ ξ = 1}.
(2.1)
In 1919, a German Mathematician Hausdorff [12] proved the convexity of W (A). If λ ∈ C does not belong to W (A), then the separation theorem implies that there exist 0 ≤ θ < 2π and b ∈ R satisfying
