From 15bfb7730801809704c6561e20c5ca47627b2d79 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sun, 21 Feb 2021 20:53:46 +0100 Subject: Add "Gram-Schmidt method" --- latex/know/concept/gram-schmidt-method/source.md | 35 ++++++++++++++++++++++ latex/know/concept/hilbert-space/source.md | 13 ++++---- .../partial-fraction-decomposition/source.md | 12 ++++---- 3 files changed, 46 insertions(+), 14 deletions(-) create mode 100644 latex/know/concept/gram-schmidt-method/source.md (limited to 'latex') diff --git a/latex/know/concept/gram-schmidt-method/source.md b/latex/know/concept/gram-schmidt-method/source.md new file mode 100644 index 0000000..b0c7b3b --- /dev/null +++ b/latex/know/concept/gram-schmidt-method/source.md @@ -0,0 +1,35 @@ +% Gram-Schmidt method + + +# 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** +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 $\ket*{n_1}$: + + $$\begin{aligned} + \ket*{n_1} = \frac{\ket*{V_1}}{\sqrt{\braket*{V_1}{V_1}}} + \end{aligned}$$ + +2. Begin loop. Take the next non-orthonormal vector $\ket*{V_j}$, and + subtract from it its projection onto every already-processed vector: + + $$\begin{aligned} + \ket*{n_j'} = \ket*{V_j} - \ket*{n_1} \braket*{n_1}{V_j} - \ket*{n_2} \braket*{n_2}{V_j} - ... - \ket*{n_{j-1}} \braket*{n_{j-1}}{V_{j-1}} + \end{aligned}$$ + + 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*: + + $$\begin{aligned} + \ket*{n_j} = \frac{\ket*{n_j'}}{\sqrt{\braket*{n_j'}{n_j'}}} + \end{aligned}$$ + +4. Loop back to step 2, taking the next vector $\ket*{V_{j+1}}$. diff --git a/latex/know/concept/hilbert-space/source.md b/latex/know/concept/hilbert-space/source.md index 7d2ea05..3f6ceb5 100644 --- a/latex/know/concept/hilbert-space/source.md +++ b/latex/know/concept/hilbert-space/source.md @@ -41,12 +41,11 @@ other. Otherwise, they would be **linearly dependent**. A vector space $\mathbb{V}$ has **dimension** $N$ if only up to $N$ of its vectors can be linearly indepedent. All other vectors in -$\mathbb{V}$ can then be written as a **linear combination** of these $N$ -so-called **basis vectors**. +$\mathbb{V}$ can then be written as a **linear combination** of these $N$ **basis vectors**. Let $\vu{e}_1, ..., \vu{e}_N$ be the basis vectors, then any vector $V$ in the same space can be **expanded** in the basis according to -the unique "weights" $v_n$, known as the **components** of the vector $V$ +the unique weights $v_n$, known as the **components** of $V$ in that basis: $$\begin{aligned} @@ -89,9 +88,9 @@ Two vectors $U$ and $V$ are **orthogonal** if their inner product $\braket{U}{V} = 0$. If in addition to being orthogonal, $|U| = 1$ and $|V| = 1$, then $U$ and $V$ are known as **orthonormal** vectors. -Orthonormality is a desirable property for basis vectors, so if they are +Orthonormality is desirable for basis vectors, so if they are not already orthonormal, it is common to manually derive a new -orthonormal basis from them using e.g. the Gram-Schmidt method. +orthonormal basis from them using e.g. the [Gram-Schmidt method](/know/concept/gram-schmidt-method). As for the implementation of the inner product, it is given by: @@ -157,7 +156,7 @@ The concept of orthonormality must be also weakened. A finite function $f(x)$ can be normalized as usual, but the basis vectors $x$ themselves cannot, since each represents an infinitesimal section of the real line. -The rationale in this case is that the identity operator $\hat{I}$ must +The rationale in this case is that action of the identity operator $\hat{I}$ must be preserved, which is given here in [Dirac notation](/know/concept/dirac-notation/): $$\begin{aligned} @@ -172,7 +171,7 @@ $$\begin{aligned} = \int_a^b \braket{x}{\xi} f(\xi) \dd{\xi} \end{aligned}$$ -For the latter integral to turn into $f(x)$, it is clear that +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/), i.e $\braket{x}{\xi} = \delta(x - \xi)$: diff --git a/latex/know/concept/partial-fraction-decomposition/source.md b/latex/know/concept/partial-fraction-decomposition/source.md index aa03f9c..69428e7 100644 --- a/latex/know/concept/partial-fraction-decomposition/source.md +++ b/latex/know/concept/partial-fraction-decomposition/source.md @@ -3,7 +3,7 @@ # Partial fraction decomposition -*Partial fraction decomposition* or *expansion* is a method to rewrite a +**Partial fraction decomposition** or **expansion** is a method to rewrite a quotient of two polynomials $g(x)$ and $h(x)$, where the numerator $g(x)$ is of lower order than $h(x)$, as a sum of fractions with $x$ in the denominator: @@ -21,9 +21,9 @@ $$\begin{aligned} } \end{aligned}$$ -Then the constant coefficients $c_n$ can either be found the hard way, +The constants $c_n$ can either be found the hard way, by multiplying the denominators around and solving a system of $N$ -equations, or the easy way by using the following trick: +equations, or the easy way by using this trick: $$\begin{aligned} \boxed{ @@ -31,8 +31,7 @@ $$\begin{aligned} } \end{aligned}$$ -If $h_1$ is a root with multiplicity $m > 1$, then the sum takes the -form of: +If $h_1$ is a root with multiplicity $m > 1$, then the sum takes the form of: $$\begin{aligned} \boxed{ @@ -41,8 +40,7 @@ $$\begin{aligned} } \end{aligned}$$ -Where $c_{1,j}$ are found by putting the terms on a common denominator, -e.g.: +Where $c_{1,j}$ are found by putting the terms on a common denominator, e.g. $$\begin{aligned} \frac{c_{1,1}}{x - h_1} + \frac{c_{1,2}}{(x - h_1)^2} -- cgit v1.2.3