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Semi-orthogonal matrix

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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

A T A = I  or  A A T = I . {\displaystyle A^{\operatorname {T} }A=I{\text{ or }}AA^{\operatorname {T} }=I.\,}

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

A T A = I n ,  and {\displaystyle A^{\operatorname {T} }A=I_{n},{\text{ and}}}
A A T = the matrix of the orthogonal projection onto the column space of  A . {\displaystyle AA^{\operatorname {T} }={\text{the matrix of the orthogonal projection onto the column space of }}A.}

The fact that A T A = I n {\textstyle A^{\operatorname {T} }A=I_{n}} implies the isometry property

‖ A x ‖ 2 = ‖ x ‖ 2 {\displaystyle \|Ax\|_{2}=\|x\|_{2}\,} for all x in R.

For example, [ 1 0 ] {\displaystyle {\begin{bmatrix}1\\0\end{bmatrix}}} is a semi-orthogonal matrix.

A semi-orthogonal matrix A is semi-unitary (either AA = 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

  1. Abadir, K.M., Magnus, J.R. (2005). Matrix Algebra. Cambridge University Press.
  2. Zhang, Xian-Da. (2017). Matrix analysis and applications. Cambridge University Press.
  3. Povey, Daniel, et al. (2018). "Semi-Orthogonal Low-Rank Matrix Factorization for Deep Neural Networks." Interspeech.


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