semi-orthogonal matrix

{{Short description|Linear algebra concept}}

{{refimprove|date=February 2014}}

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

:$A^\{\backslash operatorname\{T\}\}\; A\; =\; I\; \backslash text\{\; or\; \}\; A\; A^\{\backslash operatorname\{T\}\}\; =\; I.\; \backslash ,$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). [http://dx.doi.org/10.21437/Interspeech.2018-1417 "Semi-Orthogonal Low-Rank Matrix Factorization for Deep Neural Networks."] Interspeech.

In the following, consider the case where A is an m × n matrix for m > n.

Then

:$A^\{\backslash operatorname\{T\}\}\; A\; =\; I\_n,\; \backslash text\{\; and\}$

:$A\; A^\{\backslash operatorname\{T\}\}\; =\; \backslash text\{the\; matrix\; of\; the\; orthogonal\; projection\; onto\; the\; column\; space\; of\; \}\; A.$

The fact that $A^\{\backslash operatorname\{T\}\}\; A\; =\; I\_n$ implies the isometry property

:$\backslash |A\; x\backslash |\_2\; =\; \backslash |x\backslash |\_2\; \backslash ,$ for all x in Rⁿ.

For example, $\backslash begin\{bmatrix\}1\; \backslash \backslash \; 0\backslash end\{bmatrix\}$ 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.