論矩陣的一種變換

<正> 在本文中,數域限定為複數域.我們要來研究如下的變換:(1)(它將方陣A變成方陣B),式中P表示任意正則陣,P表示P的元素的共軛救構成的陣.所有的變换(1)顯然成羣.這種變換現在姑稱之為種變換.如果二方陣A與B可由一個種變換變此成彼,我們就說,A與B是對相似的.

ON A KIND OF TRANSFORMATIONS OF MATRICES

In this paper scalar quantities are complex numbers. The transformations studied in this paper are those of the form A→B = PAP~(-1), carrying a square matrix A into a square matrix B, where P is any non-singular matrix and P is formed of the conjugate imagineries of the elements of P. Any such transformation will be called a transformation. If two square matrices A and B can be transformed into each other by a transformation, they are said to be similar. We give an enumeration of the results arrived at in this paper. 1°Any square matrix A can be transformed by means of transformations into a certain canonical form which is uniquely determined by A.2°The similarity of the matrices is a necessary and sufficient condition for the similarity of two square matrices A and B; the same may be said of the similarity of the matrices3°Necessary and sufficient conditions (in terms of elementary divisors) are found for a square matrix to be(i) expressible in the form AA,(ii) similar to a matrix of the form(ii) similar to a matrix of the form 4°Any square matrix is similar to a real matrix.5°Any square matrix can be expressed as a product HS (or SH), where H is a Hermitian matrix and S is a symmetric matrix, a pre-assigned one of which is non-singular.We have also studied the natural extension of transformations, namely transformations of the following form: A_1 → B_1 = PA_1Q, A_2 → B_2 = PA_2, (P and Q non-singular) carrying a matric pair (A_1, A_2 ) into another matric pair (B_1, B_2). The results will be published in another paper.