It is proved that there exists a matrix (over an appropriate field) with prescribed eigenvalues and a prescribed nonprincipal diagonal. It is also proved that there exists an n x n matrix, over an arbitrary field, with prescribed eigenvalues and with n-\ prescribed arbitrary entries.

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By "multiplicative inverse eigenvalue problem" (m.i.e.p., for short) we mean the following. Let A be an n×n matrix and let s1,…, sn be n given numbers. Under what conditions does there exist an n×n diagonal matrix V such that VA has eigenvalues s1,…,sn?In the "additive inverse eigenvalue problem" (a.i.e.p., for short) we seek the diagonal matrix V so that A + V has eigenvalues s1,…, sn?.In the present paper we extend to the m.i.e.p. the ideas used in [7] for the a.i.e.p.By per X we denote the permanent of the square matrix X.

We shall be concerned with the following problem. Let a11, …, ann be complex numbers and λ1, …, λn nonnegative real numbers. Under what conditions does there exist an n × n complex matrix A with a11, …, ann as principal elements and λ1, …, λn as singular values? This problem has been suggested in [3] but, to our knowledge, has not yet been solved.