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

${\displaystyle A^{\operatorname {T} }A=I{\text{ or }}AA^{\operatorname {T} }=I.\,}$^[1][2][3]

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

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

${\displaystyle AA^{\operatorname {T} }={\text{the matrix of the orthogonal projection onto the column space of }}A.}$

The fact that ${\textstyle A^{\operatorname {T} }A=I_{n}}$ 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.