en>Sumone10154 at 02:36, 3 September 2012

2012-09-03T02:36:18Z

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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

			:<math>A^T A = I \text{ or } A A^T = I. \,</math> <ref > Abadir, K.M., Magnus, J.R. (2005). Matrix Algebra. Cambridge University Press.</ref>

			In the following, consider the case where ''A'' is an ''m'' × ''n'' matrix for ''m'' > ''n''.
			Then
			:<math>A^T A = I_n, \,</math>
			which implies the isometry property
			:<math>\\|A x\\|_2 = \\|x\\|_2 \,</math> for all ''x'' in '''R'''<sup>''n''</sup>.
			For example,
			<math>\begin{bmatrix}1 \\ 0\end{bmatrix}</math>
			is a semi-orthogonal matrix.

			A semi-orthogonal matrix ''A'' is [[semi-unitary]] (either ''A''<sup>†</sup>''A'' = ''I'' or ''AA''<sup>†</sup> = ''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==
			{{Reflist}}

			[[Category:Geometric algebra]]
			[[Category:Matrices]]


			{{linear-algebra-stub}}

en>RockMagnetist: copy edit

2012-02-22T00:00:26Z

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en>Sumone10154 at 02:36, 3 September 2012

en>RockMagnetist: copy edit