Remarks on a Matrix Transformation for Linear Differential Equations

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 [l; 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 \y\ will be employed for the norm (y*y)112 of a vector y. For Af = ||Afy||, (i, j = l, ■ ■ ■ , n), the corresponding lower case bold-face letter m,will denote the jth column vector of M. In particular, if M is a nonsingular mXm matrix, and N is the unitary matrix whose sequence of column vectors tii, • • • , nn is the set of vectors obtained by applying the Gram-Schmidt orthonormalization process to the sequence of column vectors aii, ■ • ■ , mn of M, then we shall write simply N — gs[M]. A nonsingular matrix Y(x) whose column vectors are solutions of (1) will be called a fundamental matrix for (1). A matrix M(x) will be termed "bounded on A" whenever its individual elements are bounded functions on this interval.