## 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:

  Eckart, G.; Young, G., The approximation of one matrix by another of lower rank, Psychometrika, 1, 211-218 (1936) · JFM 62.1075.02  Gauss, C. F., Theoria combinationis observationum erroribus minimus obnoxiae, (Werke IV (1821), Koniglichen Gessellschaft der Wissenschaften zu Göttingen), 1-26  Golub, G. H.; Van Loan, C., An analysis of the total least squares problem, SIAM. Numer. Anal., 17, 883-893 (1980) · Zbl 0468.65011  Golub, G. H.; Van Loan, C., Matrix Computations (1983), Johns Hopkins: Johns Hopkins Baltimore · Zbl 0559.65011  Mirsky, L., Symmetric gauge functions and unitarily invariant norms, Quart. J. Math. Oxford, 11, 50-59 (1960) · Zbl 0105.01101  Ouellette, D. V., Schur complement and statistics, Linear Algebra Appl., 36, 187-295 (1981) · Zbl 0455.15012  Stewart, G. W., A nonlinear version of Gauss’s minimum variance theorem with applications to an errors-in-the-variables model, (Computer Science Technical Report TR-1263 (1983), Univ. of Maryland)  Webster, J.; Gunst, R.; Mason, R., Latent root regression analysis, Technometrics, 16, 513-522 (1974) · Zbl 0294.62081
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