Eigenvalues of matrices with prescribed entries

It is shown that there exists an /i-square matrix all whose eigenvalues and n-1 of whose entries are arbitrarily prescribed. This result generalizes a theorem of L. Mirsky. It is also shown that there exists an «-square matrix with some of its entries prescribed and with simple eigenvalues, provided that n of the nonprescribed entries lie on a diagonal or, alternatively, provided that the number of prescribed entries does not exceed In—2. A well-known result of L. Mirsky [3] states essentially that, given any 2n—1 complex numbers Ax, ■ • • , Xn, ax, • • • , an_x, there exists an nsquare matrix with eigenvalues A,, • • • , Xn and n—\ of its main diagonal entries equal to ax, • • ■ , an_x. Related results for matrices over general fields were also obtained by Farahat and Ledermann [1]. We show that the restriction of the n—\ prescribed entries to the main diagonal is unnecessary. We first investigate the conditions under which there exists a matrix with prescribed entries and simple eigenvalues. A position in a matrix in which some entries have been prescribed is said to be free, if there is no prescribed entry in that position. By a diagonal in an «-square matrix we mean a set of n positions no two of which are in the same row or in the same column; i.e., positions (i, o(i)), /=1, •• • , n, for some permutation a. Theorem 1. Let ax, • • • , ani_n be n2—nprescribed complex numbers and let (it,ft), /= 1, ■ • • , n2—n, be prescribed different positions in an n-square matrix, such that the n remaining free positions form a diagonal of the matrix. Then there exists an n-square matrix with simple eigenvalues and with the prescribed entries at in the prescribed positions (it,jt), t= 1, • • • , n2—n. The number n of free positions cannot, in general, be decreased. Received by the editors June 1, 1971. AMS 1969 subject classifications. Primary 1525.