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.
Properties Let A {\displaystyle A} be an m × n {\displaystyle m\times n} semi-orthogonal matrix.
Either A T A = I or A A T = I . {\displaystyle A^{\operatorname {T} }A=I{\text{ or }}AA^{\operatorname {T} }=I.\,}
A semi-orthogonal matrix is an isometry. This means that it preserves the norm either in row space, or column space. A semi-orthogonal matrix always has full rank. A square matrix is semi-orthogonal if and only if it is an orthogonal matrix. A real matrix is semi-orthogonal if and only if its non-zero singular values are all equal to 1. 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).
Examples
Tall matrix (sub-isometry) Consider the 3 × 2 {\displaystyle 3\times 2} matrix whose columns are orthonormal:
A = ( 1 0 0 1 0 0 ) {\displaystyle A={\begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}}}
Here, its columns are orthonormal. Therefore, it is semi-orthogonal, which is confirmed by:
A T A = ( 1 0 0 0 1 0 ) ( 1 0 0 1 0 0 ) = ( 1 0 0 1 ) = I 2 {\displaystyle A^{T}A={\begin{pmatrix}1&0&0\\0&1&0\end{pmatrix}}{\begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}=I_{2}}
Short matrix Consider the 2 × 3 {\displaystyle 2\times 3} matrix whose rows are orthonormal:
B = ( 1 0 0 0 1 0 ) {\displaystyle B={\begin{pmatrix}1&0&0\\0&1&0\end{pmatrix}}}
Here, its rows are orthonormal. Therefore, it is semi-orthogonal, which is confirmed by:
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