Semi-orthogonal matrix
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]\displaystyle{ A^{\operatorname{T}} A = I \text{ or } A A^{\operatorname{T}} = I. \, }[/math][1][2][3]
In the following, consider the case where A is an m × n matrix for m > n. Then
- [math]\displaystyle{ A^{\operatorname{T}} A = I_n, \text{ and} }[/math]
- [math]\displaystyle{ A A^{\operatorname{T}} = \text{the matrix of the orthogonal projection onto the column space of } A. }[/math]
The fact that [math]\displaystyle{ A^{\operatorname{T}} A = I_n }[/math] implies the isometry property
- [math]\displaystyle{ \|A x\|_2 = \|x\|_2 \, }[/math] for all x in Rn.
For example, [math]\displaystyle{ \begin{bmatrix}1 \\ 0\end{bmatrix} }[/math] 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
- ↑ Abadir, K.M., Magnus, J.R. (2005). Matrix Algebra. Cambridge University Press.
- ↑ Zhang, Xian-Da. (2017). Matrix analysis and applications. Cambridge University Press.
- ↑ Povey, Daniel, et al. (2018). "Semi-Orthogonal Low-Rank Matrix Factorization for Deep Neural Networks." Interspeech.
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