Semi-orthogonal matrix

In

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 }}AA^{\operatorname {T} }=I.\,

^[1]^[2]^[3]

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

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

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

\|Ax\|_{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

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.
^ Zhang, Xian-Da. (2017). Matrix analysis and applications. Cambridge University Press.
^ Povey, Daniel, et al. (2018). "Semi-Orthogonal Low-Rank Matrix Factorization for Deep Neural Networks." Interspeech.

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

[2] Zhang, Xian-Da. (2017). Matrix analysis and applications. Cambridge University Press.

[3] Povey, Daniel, et al. (2018). "Semi-Orthogonal Low-Rank Matrix Factorization for Deep Neural Networks." Interspeech.

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