Editing QR decomposition (section)

===Using the Gram–Schmidt process===
{{further|Gram–Schmidt#Numerical stability}}
Consider the [[Gram–Schmidt process]] applied to the columns of the full column rank matrix {{nowrap|<math>A = \begin{bmatrix}\mathbf{a}_1 & \cdots & \mathbf{a}_n\end{bmatrix}</math>,}} with [[inner product]] <math>\langle\mathbf{v}, \mathbf{w}\rangle = \mathbf{v}^\textsf{T} \mathbf{w}</math> (or <math>\langle\mathbf{v}, \mathbf{w}\rangle = \mathbf{v}^\dagger \mathbf{w}</math> for the complex case).

Define the [[vector projection|projection]]:
:<math>\operatorname{proj}_{\mathbf{u}}\mathbf{a} =
  \frac{\left\langle\mathbf{u}, \mathbf{a}\right\rangle}{\left\langle\mathbf{u}, \mathbf{u}\right\rangle}{\mathbf{u}}
</math>

then:
:<math>\begin{align}
  \mathbf{u}_1 &= \mathbf{a}_1, &
    \mathbf{e}_1 &= \frac{\mathbf{u}_1}{\|\mathbf{u}_1\|} \\
  \mathbf{u}_2 &= \mathbf{a}_2 - \operatorname{proj}_{\mathbf{u}_1} \mathbf{a}_2, &
    \mathbf{e}_2 &= \frac{\mathbf{u}_2}{\|\mathbf{u}_2\|} \\
  \mathbf{u}_3 &= \mathbf{a}_3 - \operatorname{proj}_{\mathbf{u}_1} \mathbf{a}_3 - \operatorname{proj}_{\mathbf{u}_2} \mathbf{a}_3, &
    \mathbf{e}_3 &= \frac{\mathbf{u}_3}{\|\mathbf{u}_3\|} \\
                 & \;\; \vdots &  & \;\; \vdots \\
  \mathbf{u}_k &= \mathbf{a}_k - \sum_{j=1}^{k-1}\operatorname{proj}_{\mathbf{u}_j} \mathbf{a}_k,&
    \mathbf{e}_k &= \frac{\mathbf{u}_k}{\|\mathbf{u}_k\|}
\end{align}</math>

We can now express the <math>\mathbf{a}_i</math>s over our newly computed orthonormal basis:
:<math>\begin{align}
   \mathbf{a}_1 &= \left\langle\mathbf{e}_1, \mathbf{a}_1\right\rangle \mathbf{e}_1 \\
   \mathbf{a}_2 &= \left\langle\mathbf{e}_1, \mathbf{a}_2\right\rangle \mathbf{e}_1
                 + \left\langle\mathbf{e}_2, \mathbf{a}_2\right\rangle \mathbf{e}_2 \\
   \mathbf{a}_3 &= \left\langle\mathbf{e}_1, \mathbf{a}_3\right\rangle \mathbf{e}_1
                 + \left\langle\mathbf{e}_2, \mathbf{a}_3\right\rangle \mathbf{e}_2
                 + \left\langle\mathbf{e}_3, \mathbf{a}_3\right\rangle \mathbf{e}_3 \\
                &\;\;\vdots \\
   \mathbf{a}_k &= \sum_{j=1}^k \left\langle \mathbf{e}_j, \mathbf{a}_k \right\rangle \mathbf{e}_j
\end{align}</math>

where {{nowrap|<math>\left\langle\mathbf{e}_i, \mathbf{a}_i\right\rangle = \left\|\mathbf{u}_i\right\|</math>.}} This can be written in matrix form:
:<math>A = QR</math>

where:
:<math>Q = \begin{bmatrix}\mathbf{e}_1 & \cdots & \mathbf{e}_n\end{bmatrix}</math>

and
:<math>R = \begin{bmatrix}
  \langle\mathbf{e}_1, \mathbf{a}_1\rangle &
  \langle\mathbf{e}_1, \mathbf{a}_2\rangle &
  \langle\mathbf{e}_1, \mathbf{a}_3\rangle & 
                                    \cdots &
  \langle\mathbf{e}_1, \mathbf{a}_n\rangle \\
                                         0 &
  \langle\mathbf{e}_2, \mathbf{a}_2\rangle &
  \langle\mathbf{e}_2, \mathbf{a}_3\rangle & 
                                   \cdots &
  \langle\mathbf{e}_2, \mathbf{a}_n\rangle \\
                                         0 &
                                         0 &
  \langle\mathbf{e}_3, \mathbf{a}_3\rangle & 
                                    \cdots & 
  \langle\mathbf{e}_3, \mathbf{a}_n\rangle \\
                                    \vdots &
                                    \vdots &
                                    \vdots &
                                    \ddots &
                                    \vdots \\
                                         0 &
                                         0 &
                                         0 &
                                    \cdots &
 \langle\mathbf{e}_n, \mathbf{a}_n\rangle \\
                                    
\end{bmatrix}.</math>

====Example====
Consider the decomposition of
: <math>A = \begin{bmatrix}
  12 & -51 &   4 \\
   6 & 167 & -68 \\
  -4 &  24 & -41
\end{bmatrix}.</math>

Recall that an orthonormal matrix <math>Q</math> has the property {{nowrap|<math>Q^\textsf{T} Q = I</math>.}}

Then, we can calculate <math>Q</math> by means of Gram–Schmidt as follows:
: <math>\begin{align}
  U = \begin{bmatrix} \mathbf u_1 & \mathbf u_2 & \mathbf u_3 \end{bmatrix}
   &= \begin{bmatrix}
        12 & -69 & -58/5 \\
         6 & 158 &   6/5 \\
        -4 &  30 & -33
      \end{bmatrix}; \\
  Q = \begin{bmatrix}
        \frac{\mathbf u_1}{\|\mathbf u_1\|} &
        \frac{\mathbf u_2}{\|\mathbf u_2\|} &
        \frac{\mathbf u_3}{\|\mathbf u_3\|}
      \end{bmatrix} &=
      \begin{bmatrix}
         6/7 & -69/175 & -58/175 \\
         3/7 & 158/175 &   6/175 \\
        -2/7 &   6/35  & -33/35
      \end{bmatrix}.
\end{align}</math>

Thus, we have
: <math>\begin{align}
  Q^\textsf{T} A &= Q^\textsf{T}Q\,R = R; \\
               R &= Q^\textsf{T}A =
    \begin{bmatrix}
      14 &  21 & -14 \\
       0 & 175 & -70 \\
       0 &   0 &  35
    \end{bmatrix}.
\end{align}</math>

====Relation to RQ decomposition====
The RQ decomposition transforms a matrix ''A'' into the product of an upper triangular matrix ''R'' (also known as right-triangular) and an orthogonal matrix ''Q''. The only difference from QR decomposition is the order of these matrices.

QR decomposition is Gram–Schmidt orthogonalization of columns of ''A'', started from the first column.

RQ decomposition is Gram–Schmidt orthogonalization of rows of ''A'', started from the last row.

====Advantages and disadvantages====

The Gram-Schmidt process is inherently numerically unstable. While the application of the projections has an appealing geometric analogy to orthogonalization, the orthogonalization itself is prone to numerical error. A significant advantage is the ease of implementation.