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Optimal subspaces and constrained principal component analysis. (English) Zbl 1043.15002
Let $$D$$ be a linear $$d$$-dimensional subspace of the Euclidean vector space $$\mathbb R^p$$ (its elements are written as columns), and let $$z_1,z_2,\ldots,z_n$$ be points of $$\mathbb R^p$$. The author determines a linear $$d+s$$-dimensional subspace $$H$$ of $$\mathbb R^p$$ containing $$D$$ and minimizing the sum of the squared distances of the $$z_i$$’s to $$H$$. The solution is given elegantly in terms of eigenvectors which belong to the largest $$s$$ eigenvalues of the symmetric matrix $$M_DZ'ZM_D$$. Here $$M_D$$ is the matrix of the orthogonal projection of $$\mathbb R^p$$ onto $$D^\perp$$, $$Z$$ is the matrix with rows $$z_1',z_2',\ldots,z_n'$$, and a prime denotes transposition.
##### MSC:
 15A03 Vector spaces, linear dependence, rank, lineability 62H25 Factor analysis and principal components; correspondence analysis 15A18 Eigenvalues, singular values, and eigenvectors
Flury
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