Equality of Numerical Ranges of 4×4 Matrix Powers

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姓名

李佳穎(Lee Chia Ying) 查詢紙本館藏

畢業系所

數學系

論文名稱

(Equality of Numerical Ranges of 4×4 Matrix Powers)

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摘要(中)

探討當W(A) 與 W(A^k) 相等，對於所有 1 ≤ k ≤ n + 1。我們根據方陣A的unitary-similarity-invariant結構來尋找A的條件。
我們首先呈現當2×2矩陣A再一次到三次時W(A)皆相等，若且唯若A為冪等(idempotent)。則當3×3矩陣A在一次到四次時W(A)皆相等，若且唯若A么正相似(unitarily similar)於2×2冪等方正B與矩陣C的直和，且矩陣C滿足W(C^k) ⊆ W(B) 對於所有 1 ≤ k ≤ 4。我們的對於4×4矩陣的主結果將延續這個方向進行討論。

摘要(英)

In this thesis, we are interested in the question of when $W(A)$ equals $W(A^k)$ for all $1le kle n+1$. We look for conditions in terms of the unitary-similarity-invariant structure of $A$. We show that if $A$ is $2 imes 2$, then $W(A)=W(A^k)$ for all $1le kle 3$ if and only if $A$ is idempotent. We also show that if $A$ is $3 imes 3$, then $W(A)=W(A^k)$ for all $1le kle 4$ if and only if $A$ is unitarily similar to a direct sum of the form $Boplus C$, where $B$ is a $2 imes 2$ idempotent and $C$ satisfies $W(C^k)subseteq W(B)$ for all $1le kle 4$. Our main results are the analysis of $4 imes 4$ matrices along this line.

關鍵字(中)

★ 矩陣
★ 數值域

關鍵字(英)

★ Matrix
★ Numerical Ranges

論文目次

Chapter 1. Introduction................................................................1
Chapter 2. Preliminaries................................................................3
2.1 Basic properties of numerical range ................................. 3
2.2 Kippenhahn Curve..................................................4
Chapter 3. Equality of Numerical Ranges of 2 × 2 and 3 × 3 Matrices..................13
3.1 Equality of Numerical Ranges of 2 × 2 Matrices ................... 13
3.2 Equality of Numerical Ranges of 3 × 3 Matrices....................16
Chapter 4. Equality of Numerical Ranges of 4 × 4 Matrices............................26
References...............................................................44

參考文獻

[1] E. Brieskorn and H. Knorrer, Plane Algebraic Curves, Birkhauser Verlag, Basel, 1986.
[2] C.-T. Chang, H.-L. Gau, K.-Z. Wang, Equality of higher-rank numerical ranges of matrices, Linear Multilinear Algebra, 62 (2014) 626-638.
[3] J. L. Coolidge, A Treatise on Algebraic Plane Curves, Dover, New York, 1959.
[4] H.-L. Gau, K.-Z. Wang, P. Y. Wu, Euqality of Numerical Ranges of Matrix Powers, Linear Algebra Appl., to appear.
[5] H.-L. Gau, P. Y. Wu, Condition for the numerical range to contain an ellipticdisc, Linear Algebra Appl., 364 (2003) 213-222.
[6] K.-F. Lin, Numerical Ranges and Numerical Radii of Products of 3×3 Matrices, Master’s thesis, National Central University, 2015.
[7] P. R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer, New York, 1982.
[8] R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
[9] D. S. Keeler, L. Rodoman and I. M. Spitkovsky, The Numerical Range of 3 × 3 Matrices. Linear Algebra Appl., 252 (1997), 115-139.
[10] R. Kippenhahn, Uber den Wertevorrat einer Matrix, Math. Nachr., 6 (1951), 193-228.
[11] Y.-H. Liu, Elliptic Numerical Range of 4x4 matirces, Master’s thesis, National Central University, 2003.
44
[12] F. D. Murnaghan, On the field of values of a square matrix, Proc. Nat. Acad. Sci. U.S.A., 18 (1932), 246-248.
[13] S.-H. Tso, P. Y. Wu, Matrical ranges of quadratic operators, Rocky Mountain J. Math., 29 (1999) 1139-1152.
[14] P. Y. Wu, Numerical Ranges of Hilbert Space Operators, Cambridge University Press, Cambridge, to appear.

指導教授

高華隆(Hwa-Long Gau)

審核日期

2019-6-12

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