Mao-Ting Chien and Hiroshi Nakazato
Matrix Topics and Numerical Range
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Matrix Topics and Numerical Range
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Mao-Ting Chien and Hiroshi Nakazato
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About the authors Mao-Ting Chien earned the B.S. (mathematics) from Soochow University, Taiwan in 1974, the M.S. from National Taiwan University in 1976, and the Ph.D. in mathematics from State University of New York at Buffalo in 1981. Since 1981, he has been with Soochow University as an associate professor and a full professor. He served as the Chairman of Department of Mathematics, the Dean of School of Science, the Vice President for Academic Affairs, and the Dean of the Research Affairs of Soochow University. His research interest includes C*algebra, functional analysis and matrix theory. He was a Visiting Researcher at UC Santa Barbara, UC Los Angeles, University of Southern California. Email:
[email protected] Department of Mathematics, Soochow University, Taiwan Hiroshi Nakazato received a M.S. degree in mathematics at Niigata University, Japan, in 1980, and a Ph.D. degree in mathematics at Tokyo Metropolitan University in 1991. In 1986, he started his academic career at Yamagata University as an assistant. Since 1993, he has been with Hirosaki University as an associate professor and a full professor. He served as the Chairperson of the Department of Mathematical System Sciences, Hirosaki University. His research activities include matrix analysis, generalized numerical range of a matrix, linear operators in a Hilbert space, and linear operators in a Krein space. He organized the conference: Recent Development in Theory of Operators and Its Applications at Kyoto University. He also serves as a member of the editorial board of journal of International Mathematical Forum. Email:
[email protected] Department of Mathematics and Physics, Faculty of Science and Technology, Hirosaki University, Japan
Contents Preface Chapter 1 Determinant and adjugate of a matrix
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Chapter 2 Resultant and discriminant
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Chapter 3 Perturbations of matrices
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Chapter 4 Linear perturbations of Hermitian matrices
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Chapter 5 Doubly stochastic matrices and majorization
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Chapter 6 Unistochastic matrices
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Chapter 7 Eigenvalue problem of Hermitian matrices
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Chapter 8 Tridiagonalize a real symmetric matrix
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Chapter 9 Square root of a positive semi-definite matrix
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Chapter 10 Numerical range of a matrix
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Chapter 11 Numerical range plotting
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Chapter 12 Eigenvalues of the Hermitian part of a matrix
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Chapter 13 Generalized numerical ranges
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Chapter 14 Poncelet property of unitary bordering matrices
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Chapter 15 Determinantal representation of ternary forms
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References
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Exercises
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Collaboration Works
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Preface In 1996, Professor Tsuyoshi Ando organized a numerical range workshop in Sapporo. Both of the authors of this lecture notes attended the workshop, and they met there for the first time. Since then, they have long time worked together on numerical range and its related topics, and published more than 50 collaborated papers. Toeplitz(1918) proved the boundary of the numerical range of a matrix is the boundary of a convex set, and Hausdroff(1919) showed that the interior of the numerical range is connected. Due to their results, the numerical range is a convex set. The numerical range and its related subjects have been extensively studied. A biennial workshop series on the numerical range have been organized and held around the world since 1992. From the view point of algebraic curve theory, the authors of the lecture notes developed a series of works on the numerical range. This lecture notes includes some topics on matrix analysis and numerical range with emphasis on the methods of algebraic curve theory. The contents in this notes are almost self-contained This lecture notes is dedicated to the 100th anniversary of the ToeplitzHausdorff convexity theorem (1918-1919) of the numerical range. On the occasion of 20 years joint work, the authors of this notes list their collaboration papers on the theme of numerical range at the end of the lecture notes. We gratefully acknowledge the support by the Taiwan Ministry of Science and Technology, Japan Society for the Promotion of Science, and the J. T. Tai & Co Foundation Visiting Research Program for the long time collaboration research. Mao-Ting Chien
Hiroshi Nakazato May 2018
Chapter 1 Determinant and adjugate of a matrix
1.1 Matrix, Determinant, Characteristic polynomial We review some core parts of linear algebra. You learned the following simultaneous linear equations in high school: {
a11 x1 + a12 x2 = b1 a21 x1 + a22 x2 = b2
where a11 , a12 , a21 , a22 , b1 , b2 are given coefficients and x1 , x2 are unknown. For instance, { 3x + y = 11 5x + 2y = 19 We easily solve this system and find x = 3, y = 2. You learned two methods to solve the above simultaneous equations. One method is based on the substitution of the solution of one equation into another equation. If we solve the above equations by the elimination method, we know that the quantity ∆ = a11 a22 − a12 a21 plays an important role to describe the solutions. If we treat the homogeneous equations by assuming that b1 = b2 = 0, the condition ∆ = 0 is necessary and sufficient for the existence of non trivial solutions (x1 , x2 ) ̸= (0, 0) of the simultaneous linear equations. If we treat a non homogeneous simultaneous linear equations where (b1 , b2 ) ̸= (0, 0), we find that the condition ∆ ̸= 0 is necessary and sufficient for the existence of the unique solution of the equations. The condition ∆ = 0 plays an important role in the elimination of an indeterminate. We shall define an object which is a generalization of the quantity ∆ = a11 a22 − a12 a21 . Most contents of this lecture notes are based on the materials in [19] which is helpful for the study of the numerical range and its generalizations. For non-negative integers n, m, we consider an m × n matrix A = (aij ) : 1
i = 1, 2, . . . , m, j = 1, 2, . . . , n with entries aij in a commutative field K with characteristic 0. In the case n = m, the n column vectors of the matrix n × n matrix A are linearly dependent, that is, there is a non-zero column vector (x1 , x2 , . . . , xn )T ∈ K n satisfying the condition ai1 x1 + ai2 x2 + · · · + ain xn = 0, (i = 1, 2, . . . , n) if and only if the determinant of the matrix (aij ) defined by ∑ det(A) = sgnσ a1σ(1) a2σ(2) · · · anσ(n) = 0, (1.1) σ∈Sn
vanishes. This homogeneous polynomial det(A) of a11 , . . . , ann is called determinant by A. L. Cauchy in 1812. In the paper, Cauchy treated the Cauchy-Binet formula. The determinant was also studied independently by mathematicians in East Asia(cf. [39]). We consider the signatures appeared in the formula (1.1). Let Sn be the symmetric group of order n, that is, the set of all bijections of the finite set {1, 2, . . . , n} onto itself. A special type permutation ρ ∈ Sn is called a transposition if there exists a pair 1 ≤ i < j ≤ n for which ρ(i) = j, ρ(j) = i and ρ(k) = k for k ̸= i, k ̸= j. By using transposition, we explain the signature of a permutation σ ∈ Sn . Let N (σ) be the number of pair of integers 1 ≤ i < j ≤ n satisfying σ(j) < σ(i). Let η be the inverse transformation of σ. Then the number N (σ) is also characterized by the number of pairs of integers i < j satisfying η(j) < η(i).
Example 1 Let n = 5, (σ(1), σ(2), σ(3), σ(4), σ(5)) = (3, 5, 4, 1, 2), η(1) = 4, η(2) = 5, η(3) = 1, η(4) = 3, η(5) = 2. Then we exchange some consecutive two entries three times (3, 5, 4, 1, 2) → (3, 5, 1, 4, 2) → (3, 1, 5, 4, 2) → (1, 3, 5, 4, 2). Then the entry 1 reached the left end of the sequence. Similarly, the entry 2 reached the second position after three times exchanges of consecutive two entries: (1, 3, 5, 4, 2) → (1, 3, 5, 2, 4) → (1, 3, 2, 5, 4) → (1, 2, 3, 5, 4).
Example 2 Let n = 5, (σ(1), σ(2), σ(3), σ(4), σ(5)) = (3, 5, 2, 4, 1), η(1) = 5, η(2) = 3, η(3) = 1, η(4) = 4, η(5) = 2. After three times exchanges of two consecutive two entries, the entry 1 reached the left end: (3, 5, 2, 4, 1) → (3, 5, 2, 1, 4) → (3, 5, 1, 2, 4) → (3, 1, 5, 2, 4) → (1, 3, 5, 2, 4).
2
Then the entry 2 reaches the second position after two exchanges of consecutive two entries: (1, 3, 5, 2, 4) → (1, 3, 2, 5, 4) → (1, 2, 3, 5, 4). In these two examples, the positions of 1 and 2 in the sequence (σ(1), σ(2), . . . , σ(n)) are η(1)-th and η(2)-th. In Example 1, we have the inequality η(1) < η(2). In Example 2, we have η(2) < η(1). As we recognize from these two examples, the number of exchange of consecutive two entries for the entry 1 to reach the left end is η(1) − 1. This number is equal the number of all pair of natural numbers i < j satisfying σ(j) = 1 < σ(i). After the entry 1 reaches the left end, the number of exchanges of two consecutive two entries for the entry 2 reaching the second position, is the number of all pairs of natural numbers i < j satisfying σ(i) = 2 < σ(i). By continuing similar processes, we find that the sequence (σ(1), σ(2), . . . , σ(n)) is changed to (1, 2, 3, . . . , n) by N (σ) times exchanges of two consecutive entries. We find that similar effect occurs on the signature of the product ∏ (xi − xj ) 1≤i
