Editing Gram–Schmidt process (section)

== Via Gaussian elimination ==

If the rows {{math|{'''v'''<sub>1</sub>, ..., '''v'''<sub>''k''</sub>}<nowiki/>}} are written as a matrix <math>A</math>, then applying [[Gaussian elimination]] to the augmented matrix <math>\left[A A^\mathsf{T} | A \right]</math> will produce the orthogonalized vectors in place of <math>A</math>. However the matrix <math>A A^\mathsf{T}</math> must be brought to [[row echelon form]], using only the [[Elementary matrix|row operation]] of adding a scalar multiple of one row to another.<ref>{{cite journal|last1=Pursell|first1=Lyle|last2=Trimble|first2=S. Y.|title=Gram-Schmidt Orthogonalization by Gauss Elimination |journal=The American Mathematical Monthly|date=1 January 1991|volume=98|issue=6|pages=544–549| doi=10.2307/2324877 |jstor=2324877}}</ref> For example, taking <math>\mathbf{v}_1 = \begin{bmatrix} 3 & 1\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}2 & 2\end{bmatrix}</math> as above, we have
<math display="block">\left[A A^\mathsf{T} | A \right] = \left[\begin{array}{rr|rr} 10 & 8 & 3 & 1 \\ 8 & 8 & 2 & 2\end{array}\right]</math>

And reducing this to [[row echelon form]] produces
<math display="block">\left[\begin{array}{rr|rr} 1 & .8 & .3 & .1 \\ 0 & 1 & -.25 & .75\end{array}\right]</math>

The normalized vectors are then
<math display="block">\mathbf{e}_1 = \frac{1}{\sqrt {.3^2+.1^2}}\begin{bmatrix}.3 & .1\end{bmatrix} = \frac{1}{\sqrt{10}} \begin{bmatrix}3 & 1\end{bmatrix}</math>
<math display="block">\mathbf{e}_2 = \frac{1}{\sqrt{.25^2+.75^2}} \begin{bmatrix}-.25 & .75\end{bmatrix} = \frac{1}{\sqrt{10}} \begin{bmatrix}-1 & 3\end{bmatrix}, </math>
as in the example above.