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+---
+title: "Gram-Schmidt method"
+date: 2021-02-22
+categories:
+- Mathematics
+- Algorithms
+layout: "concept"
+---
+
+Given a set of linearly independent non-orthonormal vectors
+$\ket{V_1}, \ket{V_2}, ...$ from a [Hilbert space](/know/concept/hilbert-space/),
+the **Gram-Schmidt method**
+turns them into an orthonormal set $\ket{n_1}, \ket{n_2}, ...$ as follows:
+
+1.  Take the first vector $\ket{V_1}$ and normalize it to get $\ket{n_1}$:
+    
+    $$\begin{aligned}
+        \ket{n_1} = \frac{\ket{V_1}}{\sqrt{\inprod{V_1}{V_1}}}
+    \end{aligned}$$
+    
+2.  Begin loop. Take the next non-orthonormal vector $\ket{V_j}$, and
+    subtract from it its projection onto every already-processed vector:
+    
+    $$\begin{aligned}
+        \ket{n_j'} = \ket{V_j} - \ket{n_1} \inprod{n_1}{V_j} - \ket{n_2} \inprod{n_2}{V_j} - ... - \ket{n_{j-1}} \inprod{n_{j-1}}{V_{j-1}}
+    \end{aligned}$$
+    
+    This leaves only the part of $\ket{V_j}$ which is orthogonal to
+    $\ket{n_1}$, $\ket{n_2}$, etc. This why the input vectors must be
+    linearly independent; otherwise $\Ket{n_j'}$ may become zero at some
+    point.
+    
+3.  Normalize the resulting ortho*gonal* vector $\ket{n_j'}$ to make it
+    ortho*normal*:
+    
+    $$\begin{aligned}
+                \ket{n_j} = \frac{\ket{n_j'}}{\sqrt{\inprod{n_j'}{n_j'}}}
+    \end{aligned}$$
+    
+4.  Loop back to step 2, taking the next vector $\ket{V_{j+1}}$.
+
+If you are unfamiliar with this notation, take a look at [Dirac notation](/know/concept/dirac-notation/).
+
+
+
+## References
+1.  R. Shankar,
+    *Principles of quantum mechanics*, 2nd edition,
+    Springer.
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