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
In the following, consider the case where "A" is an "m" × "n" matrix for "m" > "n".
Then
The fact that formula_3 implies the isometry property
For example, formula_5 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.