Symmetric Matrices with Prescribed Eigenvalues and Eigenvectors

Abstract

It is a well-known result that if S is a real n X n symmetric matrix then it is diagonalizable, it has n mutually orthogonal eigenvectors, and there exists an orthogonal matrix P such that its columns are eigenvectors of S and such that PTSP = D is the diagonal matrix of corresponding eigenvalues [1, 2, 3, 4, and 5]. In matrix theory courses it is customary to look at a particular real n X n symmetric matrix for some small n > 2 and to try to find its eigenvalues and corresponding eigenvectors. Naturally it is up to an instructor to provide the students with nice symmetric matrices, i.e., matrices possessing integer (or rational) eigenvalues and eigenvectors with integer (or rational) components before normalization. One of the goals of this note is to show how to start with an arbitrary orthogonal basis {P 1'P2 * , Pn } of R and find a real n X n symmetric matrix having the pi for eigenvectors and having either nice eigenvalues or having n arbitrarily prescribed real numbers X I, X 2 1 . . . for its eigenvalues. Recall that a real n X n symmetric matrix S is positive (nonnegative) definite if and only if all of its eigenvectors are positive (nonnegative). It is a well-known result that a matrix S is symmetric positive definite if and only if there exists a nonsingular matrix B such that S = BBT [1, 3, and 5]. Another goal here is to sharpen this result and generalize it to characterize real symmetric matrices (Theorems 2 and 3). Our last goal is to provide a simple method for constructing nice symmetric matrices. We first present our theorems and then give several examples which illustrate the ideas and methods from the theorems and their proofs. Our first theorem will be useful in two ways; it gives an explicit method for constructing a symmetric matrix with prescribed eigenvectors and eigenvalues and it also is useful in establishing further results. Throughout our discussion, vectors will be considered as column vectors, and we will denote a matrix A with column vectors ai as A = (a1 I a2l ... I a,,).