Applications of matrix theory to approximation theory

In this paper, we examine the relationship between the eigenvalues of the arbitrary sum of n rank 1 matrices and the eigenvalues of the summands. For n = 2. 3. and 4. We develop a polynomial, p, which annihilates the matrix which is the sum of n rank 1 matrices. Since the minimal polynomial is a factor of this annihilator polynomial, then the eigenvalues of this sum are roots of p. We conjecture that his form we creatcnl works for arbitrary n. Any rank 1 matrix may be written as the product of a column vector and a row vector. The coefficients of the polynomial we created are formed using only combinations of these rows and columns. We examine conditions which imply that our annihilator polynomial is exactly the characteristic polynomial of the sum. This gives the desired relationship between the eigenvalues of the individual rank 1 matrices and the eigenvalues of the sum.