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In linear algebra, a semi-orthogonal matrix is a non-square matrix with real entries where: if the number of columns exceeds the number of rows, then the rows are orthonormal vectors; but if the number of rows exceeds the number of columns, then the columns are orthonormal vectors. Equivalently, a non-square matrix ''A'' is semi-orthogonal if either : 〔 Abadir, K.M., Magnus, J.R. (2005). Matrix Algebra. Cambridge University Press.〕 In the following, consider the case where ''A'' is an ''m'' × ''n'' matrix for ''m'' > ''n''. Then : which implies the isometry property : for all ''x'' in R''n''. For example, is a semi-orthogonal matrix. A semi-orthogonal matrix ''A'' is semi-unitary (either ''A''†''A'' = ''I'' or ''AA''† = ''I'') and either left-invertible or right-invertible (left-invertible if it has more rows than columns, otherwise right invertible). As a linear transformation applied from the left, a semi-orthogonal matrix with more rows than columns preserves the dot product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation or reflection. ==References== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Semi-orthogonal matrix」の詳細全文を読む スポンサード リンク
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