linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...

, a semi-orthogonal matrix is a non-

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

matrix with

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

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^ A = I \text A A^ = I. \,

Povey, Daniel, et al. (2018)
"Semi-Orthogonal Low-Rank Matrix Factorization for Deep Neural Networks."
Interspeech. In the following, consider the case where ''A'' is an ''m'' × ''n'' matrix for ''m'' > ''n''. Then :

A^ A = I_n, \text

A A^ = \text A.

The fact that

A^ A = I_n

implies the

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...

property :

\, A x\, _2 = \, x\, _2 \,

for all ''x'' in R^''n''. For example,

\begin1 \\ 0\end

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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

applied from the left, a semi-orthogonal matrix with more rows than columns preserves the

dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...

of vectors, and therefore acts as an isometry of

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

, such as a rotation or

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

References

Geometric algebra Matrices {{linear-algebra-stub