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

\( A^T A = I \text{ or } A A^T = I. \, \) [1]

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

\( A^T A = I_n, \,\)

which 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.

References

Abadir, K.M., Magnus, J.R. (2005). Matrix Algebra. Cambridge University Press.

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