Editing Gram–Schmidt process (section)

== Determinant formula ==
The result of the Gram–Schmidt process may be expressed in a non-recursive formula using [[determinant]]s.

<math display="block"> \mathbf{e}_j = \frac{1}{\sqrt{D_{j-1} D_j}} \begin{vmatrix}
\langle \mathbf{v}_1, \mathbf{v}_1 \rangle     & \langle \mathbf{v}_2, \mathbf{v}_1 \rangle     & \cdots & \langle \mathbf{v}_j, \mathbf{v}_1 \rangle \\
\langle \mathbf{v}_1, \mathbf{v}_2 \rangle     & \langle \mathbf{v}_2, \mathbf{v}_2 \rangle     & \cdots & \langle \mathbf{v}_j, \mathbf{v}_2 \rangle \\
\vdots                                         & \vdots                                         & \ddots & \vdots \\
\langle \mathbf{v}_1, \mathbf{v}_{j-1} \rangle & \langle \mathbf{v}_2, \mathbf{v}_{j-1} \rangle & \cdots & \langle \mathbf{v}_j, \mathbf{v}_{j-1} \rangle \\
\mathbf{v}_1                                   & \mathbf{v}_2                                   & \cdots & \mathbf{v}_j
\end{vmatrix} </math>

<math display="block"> \mathbf{u}_j = \frac{1}{D_{j-1} } \begin{vmatrix}
\langle \mathbf{v}_1, \mathbf{v}_1 \rangle     & \langle \mathbf{v}_2, \mathbf{v}_1 \rangle     & \cdots & \langle \mathbf{v}_j, \mathbf{v}_1 \rangle \\
\langle \mathbf{v}_1, \mathbf{v}_2 \rangle     & \langle \mathbf{v}_2, \mathbf{v}_2 \rangle     & \cdots & \langle \mathbf{v}_j, \mathbf{v}_2 \rangle \\
\vdots                                         & \vdots                                         & \ddots & \vdots \\
\langle \mathbf{v}_1, \mathbf{v}_{j-1} \rangle & \langle \mathbf{v}_2, \mathbf{v}_{j-1} \rangle & \cdots & \langle \mathbf{v}_j, \mathbf{v}_{j-1} \rangle \\
\mathbf{v}_1                                   & \mathbf{v}_2                                   & \cdots & \mathbf{v}_j
\end{vmatrix} </math>

where <math>D_0 = 1</math> and, for <math>j \ge 1</math>, <math>D_j</math> is the [[Gram determinant]]

<math display="block"> D_j = \begin{vmatrix}
\langle \mathbf{v}_1, \mathbf{v}_1 \rangle & \langle \mathbf{v}_2, \mathbf{v}_1 \rangle & \cdots & \langle \mathbf{v}_j, \mathbf{v}_1 \rangle \\
\langle \mathbf{v}_1, \mathbf{v}_2 \rangle & \langle \mathbf{v}_2, \mathbf{v}_2 \rangle & \cdots & \langle \mathbf{v}_j, \mathbf{v}_2 \rangle \\
\vdots & \vdots & \ddots & \vdots \\
\langle \mathbf{v}_1, \mathbf{v}_j \rangle & \langle \mathbf{v}_2, \mathbf{v}_j \rangle & \cdots & \langle \mathbf{v}_j, \mathbf{v}_j \rangle
\end{vmatrix}.  </math>

Note that the expression for <math>\mathbf{u}_k</math> is a "formal" determinant, i.e. the matrix contains both scalars and vectors; the meaning of this expression is defined to be the result of a [[Laplace expansion|cofactor expansion]] along the row of vectors.

The determinant formula for the Gram-Schmidt is computationally (exponentially) slower than the recursive algorithms described above; it is mainly of theoretical interest.