交換子與交換矩陣

Abstract

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.