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Proof: the value of a given bilinear form cannot change due to a change of basis, since the bilinear form is just a function, and does not depend on the choice of basis. The only thing that change is the matrix representation of the form. Therefore, we must have:

x^{T} M y = x_{n e w}^{T} M_{n e w} y_{n e w}

(3)

and in the new basis:

x = C x_{n e w} y = C y_{n e w} x_{n e w}^{T} M_{n e w} y_{n e w} = x^{T} M y = (C x_{n e w})^{T} M (C y_{n e w}) = x_{n e w}^{T} (C^{T} MC) y_{n e w}

(4)

and so since:

\forall x_{n e w}, y_{n e w} x_{n e w}^{T} M_{n e w} y_{n e w} = x_{n e w}^{T} (C^{T} MC) y_{n e w} ⟹ M_{n e w} = C^{T} MC

(5)

proofwiki.org/wiki/Matrix_of_Bilinear_Form_Under_Change_of_Basis

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Multilinear form by

Ciro Santilli 40 Updated 2025-07-16

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See form.

Analogous to a linear form, a multilinear form is a Multilinear map where the image is the underlying field of the vector space, e.g.

R^{n_{1}} \times R^{n_{2}} \times R^{n_{2}} \to R

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Symmetric bilinear map by

Ciro Santilli 40 Updated 2025-07-16

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Subcase of symmetric multilinear map:

f (x, y) = f (y, x)

(1)

Requires the two inputs

x

and

y

to be in the same vector space of course.

The most important example is the dot product, which is also a positive definite symmetric bilinear form.

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Symmetric bilinear form by

Ciro Santilli 40 Updated 2025-07-16

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symmetric bilinear maps that is also a bilinear form.

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Hermitian form by

Ciro Santilli 40 Updated 2025-07-16

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The complex number analogue of a symmetric bilinear form.

The prototypical example of it is the complex dot product.

Note that this form is neither strictly symmetric, it satisfies:

< x, y >= \overline{< y, x >}

(1)

where the over bar indicates the complex conjugate, nor is it linear for complex scalar multiplication on the second argument.

Bibliography:

mathworld.wolfram.com/HermitianForm.html

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Oxford physics course handbook by

Ciro Santilli 40 Updated 2025-07-16

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The normal navigation to them was paywalled, but the static files are served without login checks if you know their URL. One way to go about it is to search by prefix on the Wayback Machine: web.archive.org/web/*/https://www2.physics.ox.ac.uk/sites/default/files/contentblock/2011/06/03/*

The last handbooks we can find are 2020/2021, they might have move to a new more properly paywalled location after that year.

2020/2021:
2019/2020. They seem to have split the handbook up per year after some point.
2016/2017: www2.physics.ox.ac.uk/sites/default/files/2011-06-03/course_v3_pdf_80151.pdf 2022 archive: web.archive.org/web/20221229021312/https://www2.physics.ox.ac.uk/sites/default/files/2011-06-03/course_v3_pdf_80151.pdf
This older handbook had a more detailed course breakdown in terms of terms and weeks, e.g. on page 19.

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Matrix representation of a Hermitian form by

Ciro Santilli 40 Updated 2025-07-16

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;

A Hermitian matrix.

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Positive definite symmetric bilinear form by

Ciro Santilli 40 Updated 2025-07-16

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Symmetric bilinear form that is also positive definite, i.e.:

\forall x, B (x, x) > 0

(1)

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Matrix representation of a positive definite symmetric bilinear form by

Ciro Santilli 40 Updated 2025-07-16

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A positive definite matrix that is also a symmetric matrix.

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Representation theory by

Ciro Santilli 40 Updated 2025-07-16

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Basically, a "representation" means associating each group element as an invertible matrices, i.e. a matrix in (possibly some subset of)

G L (n)

, that has the same properties as the group.

Or in other words, associating to the more abstract notion of a group more concrete objects with which we are familiar (e.g. a matrix).

Each such matrix then represents one specific element of the group.

This is basically what everyone does (or should do!) when starting to study Lie groups: we start looking at matrix Lie groups, which are very concrete.

Or more precisely, mapping each group element to a linear map over some vector field

V

(which can be represented by a matrix infinite dimension), in a way that respects the group operations:

R (g) : G \to G L (V)

(1)

As shown at Physics from Symmetry by Jakob Schwichtenberg (2015)

page 51, a representation is not unique, we can even use matrices of different dimensions to represent the same group
3.6 classifies the representations of $S U (2)$ . There is only one possibility per dimension!
3.7 "The Lorentz Group O(1,3)" mentions that even for a "simple" group such as the Lorentz group, not all representations can be described in terms of matrices, and that we can construct such representations with the help of Lie group theory, and that they have fundamental physical application

Motivation:

math.stackexchange.com/questions/1628464/what-is-representation-theory

Bibliography:

www.youtube.com/watch?v=9rDzaKASMTM "RT1: Representation Theory Basics" by MathDoctorBob (2011). Too much theory, give me the motivation!
www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609 The "Useless" Perspective That Transformed Mathematics by Quanta Magazine (2020). Maybe there is something in there amidst the "the reader might not know what a matrix is" stuff.

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Skew-symmetric bilinear map by

Ciro Santilli 40 Updated 2025-07-16

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Subcase of antisymmetric multilinear map:

f (x, y) = - f (y, x)

(1)

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Skew-symmetric bilinear form by

Ciro Santilli 40 Updated 2025-07-16

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Skew-symmetric bilinear map that is also a bilinear form.

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Tesselation by

Ciro Santilli 40 Updated 2025-07-16

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B3 Oxford physics course by

Ciro Santilli 40 Updated 2025-07-16

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users.physics.ox.ac.uk/~lvovsky/B3/ contain assorted PDFs from between 2015 and 2019

Syllabus reads:

Multi-electron atoms: central field approximation, electron configurations, shell structure, residual electrostatic interaction, spin orbit coupling (fine structure).
Spectra and energy levels: Term symbols, selection rules, X-ray notation, Auger transitions.
Hyperfine structure; effects of magnetic fields on fine and hyperfine structure. Presumably Zeeman effect.
Two level system in a classical light field: Rabi oscillations and Ramsey fringes, decaying states; Einstein
A and B coefficients; homogeneous and inhomogeneous broadening of spectral lines; rate equations.
Optical absorption and gain: population inversion in 3- and 4-level systems; optical gain cross section; saturated absorption and gain.

Professor in 2000s seems to be

en.wikipedia.org/wiki/Paul_Ewart. He actually fought not to be dismissed by age and won!
www.physics.ox.ac.uk/our-people/ewart

But as of 2023 marked emeritus, so who took over?

Ewart is actually religious:

www.youtube.com/watch?v=aulL-Qa65i0 Paul Ewart, Chance, Science and Spirituality by Faraday Institute for Science and Religion. Oh, he is/was actually chairman of that crap
www.youtube.com/watch?v=PVX2F4XvGYo Chaos and the Character of God by Prof. Paul Ewart

This dude is pure trouble for Oxford!

Undated materials Ewart:

users.physics.ox.ac.uk/~ewart/index.htm
users.physics.ox.ac.uk/~ewart/Atomic%20Physics%20lecture%20notes%20C%20port.pdf
slides: users.physics.ox.ac.uk/~ewart/Atomic%20Physics%20Lecture%20PPT%20slides%201_8.pdf. Also under: www2.physics.ox.ac.uk/sites/default/files/2011-10-19/atomic_physics_lectures_1_8_09_pdf_pdf_18283.pdf. The course was previously B1, they just change the IDs randomly from time to time to fit the B1-7 numbering.

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Antisymmetric multilinear map by

Ciro Santilli 40 Updated 2025-07-16

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Change sign if you swap two input values.

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Dot product by

Ciro Santilli 40 Updated 2025-07-16

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The definition of the "dot product" of a general space varies quite a lot with different contexts.

Most definitions tend to be bilinear forms.

We use the unqualified generally refers to the dot product of Real coordinate spaces, which is a positive definite symmetric bilinear form. Other important examples include:

the complex dot product, which is not strictly symmetric nor linear, but it is positive definite
Minkowski inner product, sometimes called" "Minkowski dot product is not positive definite

The rest of this section is about the

R^{n}

case.

The positive definite part of the definition likely comes in because we are so familiar with metric spaces, which requires a positive norm in the norm induced by an inner product.

The default Euclidean space definition, we use the matrix representation of a symmetric bilinear form as the identity matrix, e.g. in

R^{3}

M = 100010001

(1)

so that:

x \cdot y = [x_{1} x_{2} x_{3}] 100010001 y_{1} y_{2} y_{3} = x_{1} y_{1} + x_{2} y_{2} + x_{3} y_{3}

(2)

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Index picking function by

Ciro Santilli 40 Updated 2025-07-16

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Levi-Civita symbol by

Ciro Santilli 40 Updated 2025-07-16

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Denoted by the Greek letter epsilon with \varepsilon encoding in LaTeX.

Definition:

odd permutation: -1
even permutation: 1
not a permutation: 0. This happens iff two more more indices are repeated

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Projection (mathematics) by

Ciro Santilli 40 Updated 2025-07-16

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 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either https://OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade