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 matrix (mathemat ...

, a semi-orthogonal matrix is a non-

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

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^ 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 pr ...

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 (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...

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, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

, such as a

rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

or reflection.

References

Geometric algebra Matrices (mathematics) {{matrix-stub