Factorization of singular matrices

Sourour, A. R.; Tang, Kunikyo

doi:10.1090/S0002-9939-1992-1097352-4

Factorization of singular matrices
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by A. R. Sourour and Kunikyo Tang

Proc. Amer. Math. Soc. 116 (1992), 629-634

DOI: https://doi.org/10.1090/S0002-9939-1992-1097352-4

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Abstract:

We give a necessary and sufficient condition that a singular square matrix $A$ over an arbitrary field can be written as a product of two matrices with prescribed eigenvalues. Except when $A$ is a $2 \times 2$ nonzero nilpotent, the condition is that the number of zeros among the eigenvalues of the factors is not less than the nullity of $A$. We use this result to prove results about products of hermitian and positive semidefinite matrices simplifying and strengthening some known results.

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