Semi-orthogonal matrix

Short description: Linear algebra concept

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 $A$ be an $m\times n$ semi-orthogonal matrix.

• Either $A^{\mathrm{T}}A=I\text{ or }A{A}^{\mathrm{T}}=I.$
• 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 $3\times 2$ matrix whose columns are orthonormal: $A=\left(\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right)$ Here, its columns are orthonormal. Therefore, it is semi-orthogonal, which is confirmed by: $A^{T}A=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=I_2$

Short matrix

Consider the $2\times 3$ matrix whose rows are orthonormal: $B=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right)$ Here, its rows are orthonormal. Therefore, it is semi-orthogonal, which is confirmed by: $B{B}^T=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=I_2$

Non-example

The following $3\times 2$ matrix has orthogonal, but not orthonormal, columns and is therefore not semi-orthogonal: $C=\left(\begin{array}{cc}2& 0\\ 0& 1\\ 0& 0\end{array}\right)$ The calculation confirms this: $C^{T}C=\left(\begin{array}{ccc}2& 0& 0\\ 0& 1& 0\end{array}\right)\left(\begin{array}{cc}2& 0\\ 0& 1\\ 0& 0\end{array}\right)=\left(\begin{array}{cc}4& 0\\ 0& 1\end{array}\right)\ne I_2$

Proofs

Preservation of Norm

If a matrix $A$ is tall or square ($m\ge n$), its semi-orthogonality implies $A^{T}A=I_n$. For any vector $x\in {\mathbb{ℝ}}^n$, $A$ preserves its norm: $\Vert Ax{\Vert }_2^2=(Ax{)}^T(Ax)=x^T{A}^{T}Ax=x^T{I}_{n}x=\Vert x{\Vert }_2^2$ If a matrix $A$ is short ($m<n$), it preserves the norm of vectors in its row space.

Justification for Full Rank

If $A^{T}A=I_n$, then the columns of $A$ are linearly independent, so the rank of $A$ must be $n$. If $A{A}^T=I_m$, then the rows of $A$ are linearly independent, so the rank of $A$ must be $m$. In both cases, the matrix has full rank.

Singular Value Property

The statement is that a real matrix $A$ is semi-orthogonal if and only if all of its non-zero singular values are 1.

This follows directly from the SVD, $A=U\mathrm{\Sigma }{V}^T$.

($⟹$) Assume $A$ is semi-orthogonal. Then either $A^{T}A=I$ or $A{A}^T=I$. The non-zero singular values of $A$ are the square roots of the non-zero eigenvalues of both $A^{T}A$ and $A{A}^T$. Since one of these "Gramian" matrices is an identity matrix, its eigenvalues are all 1. Thus, the non-zero singular values of $A$ must be 1.

($\Leftarrow$) Assume all non-zero singular values of $A$ are 1. This forces the block of $\mathrm{\Sigma }$ containing the non-zero values to be an identity matrix. This structure ensures that either ${\mathrm{\Sigma }}^T\mathrm{\Sigma }=I_n$ (if $A$ has full column rank) or $\mathrm{\Sigma }{\mathrm{\Sigma }}^T=I_m$ (if $A$ has full row rank). Substituting this into the expressions for $A^{T}A=V({\mathrm{\Sigma }}^T\mathrm{\Sigma })V^T$ or $A{A}^T=U(\mathrm{\Sigma }{\mathrm{\Sigma }}^T)U^T$ respectively shows that one of them must simplify to an identity matrix, satisfying the definition of a semi-orthogonal matrix.