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

${\displaystyle A^{T}A=I{\text{ or }}AA^{T}=I.\,}$ [1]

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

${\displaystyle A^{T}A=I_{n},\,}$

which implies the isometry property

${\displaystyle \|Ax\|_{2}=\|x\|_{2}\,}$ for all x in Rⁿ.

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