Construction of matrices with prescribed singular values and eigenvalues
نویسنده
چکیده
Two issues concerning the construction of square matrices with prescribed singular values and eigenvalues are addressed. First, a necessary and sufficient condition for the existence of an n × n complex matrix with n given nonnegative numbers as singular values and m(≤ n) given complex numbers to be m of the eigenvalues is determined. This extends the classical result of Weyl and Horn treating the case when m = n. Second, an algorithm is given to generate a triangular matrix with prescribed singular values and eigenvalues. Unlike earlier algorithms, the eigenvalues can be arranged in any prescribed order on the diagonal. A slight modification of this algorithm allows one to construct a real matrix with specified real and complex conjugate eigenvalues and specified singular values. The construction is done by multiplication by diagonal unitary matrices, permutation matrices and rotation matrices. It is numerically stable and may be useful in developing test software for numerical linear algebra packages.
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