Classical Gram-Schmidt

Orthogonalizing a set of vectors

Algorithm

We process each vector $a_j$ in turn. First, we project it onto all previously computed orthogonal vectors $q_i$ (where $i < j$ ). We subtract these projections to get a residual vector $u_j$ , which is orthogonal to all previous $q_i$ . Finally, we normalize $u_j$ to get $q_j$ .

Step-by-Step Construction

Step 0 of 5

StartFinish

Initial vectors

a_1, a_2, a_3

R Matrix Construction

R = \begin{bmatrix} \hphantom{r_{11}}\llap{0} & \hphantom{r_{12}}\llap{0} & \hphantom{r_{13}}\llap{0} \\[0.5em] \hphantom{r_{21}}\llap{0} & \hphantom{r_{22}}\llap{0} & \hphantom{r_{23}}\llap{0} \\[0.5em] \hphantom{r_{31}}\llap{0} & \hphantom{r_{32}}\llap{0} & \hphantom{r_{33}}\llap{0} \end{bmatrix}

Entries $r_{ij} = q_i^T a_j$ and $r_{jj} = \\|u_j\\|$

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