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Topics in Matrix Analysis by Charles R. Johnson, Roger A. Horn

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Chapter 3

Singular value inequalities

3.0    Introduction and historical remarks

Singular values and the singular value decomposition play an important role in high-quality statistical computations and in schemes for data compression based on approximating a given matrix with one of lower rank. They also play a central role in the theory of unitarily invariant norms. Many modern computational algorithms are based on singular value computations because the problem of computing the singular values of a general matrix (like the problem of computing the eigenvalues of a Hermitian matrix) is well-conditioned; for numerous examples see [GVl].

    There is a rich abundance of inequalities involving singular values, and a selection from among them is ...

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