標題: 交換子與交換矩陣Commutator and Commutant 作者: 潘淑真Shu-Chen Pan吳培元Pei Yuan Wu應用數學系所 關鍵字: 交換子;交換矩陣;交換;Commutator;Commutant;commute 公開日期: 2002 摘要: 在本論文中，我們探討有關矩陣交換子的性質，以及兩個矩陣的commutant的維數性質。 首先，讓A是個 n-by-n矩陣，我們証明出以下二件事是對等的，(a) A的每一個特徵值的幾何重數，不是等於1，就是等於它的代數重數。(b) 給任意的n-by-n矩陣B，如果A跟B的交換子C=AB-BA即跟A互換，又跟B 互換，則這樣的C必為零。 接下來，讓A和B是兩個n-by-n的可互換矩陣，我們証明出，如果相對於A的每一個特徵值，均有不超過兩個的Jordan block，或是每一個Jordan block 都是1-by-1的，則A和B的commutant的維數至少會是n。此外，我們也完整的指出何時等號會成立。譬如當A是nonderogatory 時，就是一個例子。In this thesis, we study properties of matrix commutators and the dimension of the commutant of two commuting matrices. First, we show that the following are equivalent conditions on a matrix A 2 Mn : (a) The geometric multiplicity of each eigenvalue of A is either equal to 1 or equal to its algebraic multiplicity. (b) For any B 2 Mn(C); if commutator C = AB ¡ BA commutes with both A and B, then C must be zero. Next, let A be an n-by-n complex matrix. If every eigenvalue of A has no more than two Jordan blocks or is associated with only 1-by-1 Jordan blocks, then for any B commutes with A, the dimension of the commutant of A and B is at least n. Moreover, under this condition on A we also completely determine when the above dimension equals n. In particular, this is the case when A is nonderogatory. URI: http://140.113.39.130/cdrfb3/record/nctu/#NT910507006http://hdl.handle.net/11536/70939 Appears in Collections: Thesis