在本篇論文中，對任意3×3的複數矩陣A和B，我們給出了充分且必要的條件對於AB矩陣乘積的數值域和BA矩陣乘積的數值域相等時。此外，去研究當A和A2的數值域半徑為1且A3的數值域半徑小於1時，A會有什麼樣的矩陣結構。以及最後，我們給出了充分且必要的條件對於當A為壓縮矩陣其特徵值長度皆小於1且A的範數為1，A與B張量積的數值域半徑等於A的範數與B的數值域半徑乘積時。

In this thesis, for any two 3-by-3 complex matrices A and B, we show that the necessary and sufficient conditions for the equality W(AB) = W(BA) to hold, where W(
) denotes the numerical range of a matrix, and the structure of A when w(A) =w (A2) = 1 and w (A3) < 1, where w(
) denotes the numerical radius of a matrix, and obtain the necessary and sufficient condition for the equality w(A B) = kAkw(B)to hold when A is a completely nonunitary contraction with kAk = 1, where k
k
denotes the usual operator norm of a matrix.

1. T. Ando, On the structure of operators with numerical radius one, Acta Sci.
Math. (Szeged) 34 (1973), 11{15.
2. C. A. Berger, A strange dilation theorem, Notices Amer. Math. Soc. 12 (1965),
590.
3. William F. Donoghue, On the numerical range of a bounded operator, Michigan
Math. J. 4 (1957), no. 3, 261{263.
4. Miroslav Fiedler, Geometry of the numerical range of matrices, Linear Algebra
and its Applications 37 (1981), 81{96.
5. Hwa-Long Gau, Chi-Kwong Li, and Pei Yuan Wu, Higher-rank numerical
ranges and dilations, Journal of Operator Theory 63 (2010), no. 1, 181.
6. Hwa-Long Gau, Kuo-Zhong Wang, and Pei Yuan Wu, Numerical radii for tensor
products of matrices, Linear and Multilinear Algebra (2013), no. ahead-of-
print, 1{21.
7. Hwa-Long Gau and Pei Yuan Wu, Finite Blaschke products of contractions,
Linear Algebra and its Applications 368 (2003), 359{370.
8. Moshe Goldberg, Eitan Tadmor, and Gideon Zwas, The numerical radius and
spectral matrices, Linear and Multilinear Algebra 2 (1975), no. 4, 317{326.
9. Karl E. Gustafson and Duggirala K. M. Rao, Numerical range: the eld of
values of linear operators and matrices, Universitext, Springer, New York, 1997.
10. Paul Richard Halmos, A Hilbert space problem book, vol. 19, Springer Science
& Business Media, 1982.
11. Felix Hausdor, Der wertvorrat einer bilinearform, Mathematische Zeitschrift
3 (1919), no. 1, 314{316.
12. Leslie Hogben, Handbook of linear algebra, 2nd ed., Discrete Mathematics and
Its Applications, Taylor and Francis, Hoboken, NJ, 2013.
13. John A. R. Holbrook, Multiplicative properties of the numerical radius in operator
theory, J. reine angew. Math 237 (1969), 166{174.
14. Roger A. Horn and Charles R. Johnson, Topics in matrix analysis, Cambridge
University Press, 1991, Cambridge Books Online.
15. Dennis S. Keeler, Leiba Rodman, and Ilya M. Spitkovsky, The numerical range
of 3  3 matrices, Linear Algebra and its Applications 252 (1997), 115{139.
16. Rudolf Kippenhahn,  Uer den wertevorrat einer matrix, Mathematische
Nachrichten 6 (1951), 193{228.
17. Yue-Hua Lu, Numerical ranges and numerical radii for tensor products of matrices
, Doctoral Dissertation (2015), National Central University.
18. Francis D. Murnaghan, On the eld of values of a square matrix, Proceedings
of the National Academy of Sciences of the United States of America 18 (1932),
no. 3, 246.
19. Dragomir Z. Dokovic and Charles R. Johnson, Unitarily achievable zero patterns
and traces of words in a and a, Linear Algebra and its Applications 421
(2007), no. 1, 63{68, Special Issue devoted to the 12th fILASg Conference 12th
Conference of the International Linear Algebra Society.
20. Carl Pearcy, An elementary proof of the power inequality for the numerical
radius., The Michigan Mathematical Journal 13 (1966), no. 3, 289{291.
21. Vlastimil Ptak, Norms and the spectral radius of matrices, Czechoslovak Math-
ematical Journal 12 (1962), no. 4, 555{557.
22. Otto Toeplitz, Das algebraische analogon zu einem satze von fejer, Mathema-
tische Zeitschrift 2 (1918), no. 1, 187{197.
23. Aurel Wintner, Zur theorie der beschrankten bilinearformen, Mathematische
Zeitschrift 30 (1929), no. 1, 228{281.
3