Remarks on a matrix transformation for linear differential equations

Abstract

The remarks of this note are concerned with a result on transformations stated below as Theorem A, and are two-fold in nature: firstly, there are comments on the relation of this theorem to results of Perron [3] and Diliberto [1; 2], in the hope of correcting a misunderstanding that has arisen in this regard; secondly, there are remarks stressing two general properties of admissible transformation matrices which together afford a very elementary matrix proof of Theorem A. Matrix notation will be used throughout, with a vector considered as a one-column matrix. If M is a matrix then the corresponding transpose and conjugate-transpose matrices are denoted by M and M*, respectively. The symbol YI will be employed for the norm (y*y)112 of a vector y. For M-IIMiAlI, (i, j=1, , n), the corresponding lower case bold-face letter mj will denote the jth column vector of M. In particular, if M is a nonsingular n Xn matrix, and N is the unitary matrix whose sequence of column vectors nli, * * *, nn is the set of vectors obtained by applying the Gram-Schmidt orthonormalization process to the sequence of column vectors mi, * , Mn of M, then we shall write simply N=gs [M]. nonsingular matrix Y(x) whose column vectors are solutions of (1) will be called a fundamental matrix for (1). matrix M(x) will be termed on A whenever its individual elements are bounded functions on this interval.