摘 要

在這篇文章中，最主要我們要討論什麼樣的矩陣A才可以滿足(5)(see this thesis)這個等式，其中total cosine的定義和operator的角度有關，而有關operator的角度的問題在過去已經有很豐富的結果，而我們現在只針對這total cosine的定義所牽涉到的角度問題來做探討。在第一章，我們介紹total sine 和total cosine的定義以及閱讀這篇文章所需的先備知識並且介紹[11]中所有2-by-2矩陣之相關結果。在第二章中，我們討論n-by-n的nonnormal矩陣A能否有 這等式，並且檢驗Minmax equality([8])，而在第三章，我們則討論n-by-n的normal矩陣是否可以滿足此等式。

Abstract

In this thesis, we consider what kind of matrices A can make the equality (5) hold? In [11], it is proved that almost 2-by-2 matrices satisfy the equality (5) and we state these results in Chapter1. In chapter2, we show that not all of the n-by-n nonnormal matrices can satisfy the equality by given an example, and this example show that the Minmax equality in [8] is false. Finally, In chapter3, we study this problem on n-by-n normal matrices.

REFERENCE
[1] E.Aspulund and V.Ptak(1971). A Minmax Inequality for Operators and a Related Numerical Range, Acta Math. 126,53-62.
[2] C.Davis. Extending the Kantorovic Inequality to Normal Matrices, Linear Algebra. Appl.31:173-177(1980).
[3]K.Gustatson. Matrix Trigonometry matrix, Lin. Alg. Appl .217:117
140 (1995).
[4] K.Gustatson. Operator products and operator angles Notices Amer. Math.Soc. 14:943 (1967).
[5] K.Gustafson. Operator trigonometry, Linear and Multilinear Algebra 37:139-159 (1994).
[6] K.Gustatson(1968d). A Min-Max Theorem, Notices Amer. Math. Soc 15,799.
[7] K.E.Gustafson(1970). The Toeplitz-Hausdorff Theorem for Linear Operators, Proc. Amer. Math. Soc. 25,203-204.
[8]Karl.E. Gustafson Duggirala K.M. Rao Numerical Range.
[9] T.I.Seidman. An Identity for normal-like Operators, Isrel J.Math.7:249-253(1969).
[10] M.H.Stone(1932). Linear Transformation in Hilbert space, American Mathematical Society, R.I.
[11] I Yin Wang. Trigonometry of two-by-two matrices. National Chen kung University 2006.