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


 * $$A^{\operatorname{T}} A = I \text{ or } A A^{\operatorname{T}} = I. \,$$

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


 * $$A^{\operatorname{T}} A = I_n, \text{ and}$$
 * $$A A^{\operatorname{T}} = \text{the matrix of the orthogonal projection onto the column space of } A.$$

The fact that $A^{\operatorname{T}} A = I_n$ implies the isometry property


 * $$\|A x\|_2 = \|x\|_2 \,$$ for all x in Rn.

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