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A generalization of the Eckart-Young-Mirsky matrix approximation theorem. (English) Zbl 0623.15020

Let X be an \(n\times p\) matrix with \(n\geq p\) and let \(\| \cdot \|\) be a unitarily invariant matrix norm. Let \(X=(X_ 1,X_ 2)\) where \(X_ 1\) has k columns. The problem considered in this paper is: find a matrix \(\hat X{}_ 2\) such that \(rank(X_ 1,\hat X_ 2)\leq r\) and \[ \| (X_ 1,\hat X_ 2)-(X_ 1,X_ 2)\| =\inf_{rank(X_ 1,\bar X_ 2)\leq \quad r}\| (X_ 1,\bar X_ 2)-(X_ 1,X_ 2)\|. \] This problem was solved by C. Eckart and G. Young [The approximation of one matrix by another of lower rank, Psychometrika 1, 211-218 (1936)] in the case \(k=0\), for the Frobenius norm. Let \(H_ r(X)\) denote the Eckart-Young solution (with \(H_ r(X)=X\) if \(r>p)\). The authors prove the following:
Theorem. Let \(X=(X_ 1,X_ 2)\) where \(X_ 1\) has k columns ad let \(\ell =rank X_ 1\). Let P denote the orthogonal projection onto the column space of X and \(P^{\perp}\) the orthogonal projection onto its orthogonal complement. If \(\ell \leq r\) then the matrix \(\hat X{}_ 2=PX_ 2+H_{r-\ell}(P^{\perp}X_ 2)\) is a solution of the problem above. A number of consequences of this theorem are considered and, in particular, applications to multiple correlations, variance inflation factors and total least squares are given.
Reviewer: F.J.Gaines

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A24 Matrix equations and identities
62H20 Measures of association (correlation, canonical correlation, etc.)
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References:

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