Semi-orthogonal matrix

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a semi-orthogonal matrix is a non-<a href="/facts/Square_matrix/5TP9mNNn">square</a> <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> with <a href="/facts/Real_number/R02gw5Pb">real</a> entries where: if the number of columns exceeds the number of rows, then the rows are <a href="/facts/Orthonormal_vectors/JfgL1nAh">orthonormal vectors</a>; 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
 
 T
 
 
 A
 =
 I
 
  or 
 
 A
 
 A
 
 T
 
 
 =
 I
 .
 
 
 
 {\displaystyle A^{\operatorname {T} }A=I{\text{ or }}AA^{\operatorname {T} }=I.\,}
 
<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
In the following, consider the case where A is an m × n matrix for m > n.
Then

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

A
        
          A
          
            T
          
        
        =
        
          the matrix of the orthogonal projection onto the column space of 
        
        A
        .
      
    
    {\displaystyle AA^{\operatorname {T} }={\text{the matrix of the orthogonal projection onto the column space of }}A.}

The fact that 
 
 
 
 
 A
 
 T
 
 
 A
 =
 
 I
 
 n
 
 
 
 
 {\textstyle A^{\operatorname {T} }A=I_{n}}
 
 implies the <a href="/facts/Isometry/K5wX8ah6">isometry</a> property

‖
 A
 x
 
 ‖
 
 2
 
 
 =
 ‖
 x
 
 ‖
 
 2
 
 
 
 
 
 {\displaystyle \|Ax\|_{2}=\|x\|_{2}\,}
 
 for all x in Rn.
For example, 
 
 
 
 
 
 [
 
 
 
 1
 
 
 
 
 0
 
 
 
 ]
 
 
 
 
 {\displaystyle {\begin{bmatrix}1\\0\end{bmatrix}}}
 
 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 <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a> applied from the left, a semi-orthogonal matrix with more rows than columns preserves the <a href="/facts/Dot_product/tNz8MLjT">dot product</a> of vectors, and therefore acts as an isometry of <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, such as a <a href="/facts/Rotation_(mathematics)/6gtkkcy0">rotation</a> or <a href="/facts/Reflection_(mathematics)/5JHWnrql">reflection</a>.

Semi-orthogonal matrix open-in-new

Semi-orthogonal matrix