diff options
Diffstat (limited to 'latex/know/concept')
| -rw-r--r-- | latex/know/concept/gram-schmidt-method/source.md | 5 | ||||
| -rw-r--r-- | latex/know/concept/hilbert-space/source.md | 8 |
2 files changed, 8 insertions, 5 deletions
diff --git a/latex/know/concept/gram-schmidt-method/source.md b/latex/know/concept/gram-schmidt-method/source.md index b0c7b3b..7920a30 100644 --- a/latex/know/concept/gram-schmidt-method/source.md +++ b/latex/know/concept/gram-schmidt-method/source.md @@ -4,7 +4,8 @@ # Gram-Schmidt method Given a set of linearly independent non-orthonormal vectors -$\ket*{V_1}, \ket*{V_2}, ...$ from a Hilbert space, the **Gram-Schmidt method** +$\ket*{V_1}, \ket*{V_2}, ...$ 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 $\ket*{n_1}$: @@ -33,3 +34,5 @@ turns them into an orthonormal set $\ket*{n_1}, \ket*{n_2}, ...$ as follows: \end{aligned}$$ 4. Loop back to step 2, taking the next vector $\ket*{V_{j+1}}$. + +If you are unfamiliar with this notation, take a look at [Dirac notation](/know/concept/dirac-notation/). diff --git a/latex/know/concept/hilbert-space/source.md b/latex/know/concept/hilbert-space/source.md index 3f6ceb5..780fc0a 100644 --- a/latex/know/concept/hilbert-space/source.md +++ b/latex/know/concept/hilbert-space/source.md @@ -89,8 +89,8 @@ $\braket{U}{V} = 0$. If in addition to being orthogonal, $|U| = 1$ and $|V| = 1$, then $U$ and $V$ are known as **orthonormal** vectors. Orthonormality is desirable for basis vectors, so if they are -not already orthonormal, it is common to manually derive a new -orthonormal basis from them using e.g. the [Gram-Schmidt method](/know/concept/gram-schmidt-method). +not already like that, it is common to manually turn them into a new +orthonormal basis using e.g. the [Gram-Schmidt method](/know/concept/gram-schmidt-method). As for the implementation of the inner product, it is given by: @@ -171,8 +171,8 @@ $$\begin{aligned} = \int_a^b \braket{x}{\xi} f(\xi) \dd{\xi} \end{aligned}$$ -For the latter integral to turn into $f(x)$, it is plain to see that -$\braket{x}{\xi}$ must be a [Dirac delta function](/know/concept/dirac-delta-function/), +Since we want the latter integral to reduce to $f(x)$, it is plain to see that +$\braket{x}{\xi}$ can only be a [Dirac delta function](/know/concept/dirac-delta-function/), i.e $\braket{x}{\xi} = \delta(x - \xi)$: $$\begin{aligned} |