7 ms·

How Lewis Carroll computed determinants (2023)

209 points by tzury 10mo ago

messe 10mo ago
HN title filter cut off the initial "How". You can manually edit it back in.
- marcusestes 10mo ago
  “Drop the ‘how.’ It’s cleaner.”
  - deleted 10mo ago
    [deleted]
  - vharuck 10mo ago
    It gives it a different implication. As I read it, an article titled "Lewis Carroll Computed Determinates" has three possible subjects: 1. Literally, Carroll would do matrix math. I know, like many on HN, that he was a mathematician. So this would be a dull and therefore unlikely subject. 2. Carroll invented determinates. This doesn't really fit the timeline of math history, so I doubt it. 3. Carroll computed determinates, and this was surprising. Maybe because we thought he was a bad mathematician, or the method had recently been invented and we don't know how he learned of it. This is slightly plausible. 4. (The actual subject). Carroll invented a method for computing determinates. A mathematician inventing a math technique makes sense, but the title doesn't. It'd be like saying "Newton and Leibnitz Used Calculus." Really burying the lede. Of course, this could've been avoided had the article not gone with a click-bait style title. A clearer one might've been "Lewis Carroll's Method for Calculating Determinates Is Probably How You First Learned to Do It." It's long, but I'm not a pithy writer. I'm sure somebody could do better.
    - SAI_Peregrinus 10mo ago
      "Lewis Carroll's Method for Computing Determinates" is headlinese and avoids the "how".
    - miltonlost 10mo ago
      "How Lewis Carroll Computed Determinates" is fine and not clickbait because it provides all the pertinent information and is an accurate summary of its contents. Clickbait would be "you would never guess how this author/mathematician computed determinants" since it requires a clickthrough to know who the person is. How is perfectly fine IMO to have in the title because I personally would expect the How to be long enough to warrant a necessary clickthrough due to the otherwise required title length.
      - 867-5309 10mo ago
        it's not quite McKean's Law so I'll settle for contagious
esafak 10mo ago
> Arrange the given block, if necessary, so that no ciphers [zeros] occur in its interior. I forgot that cipher used to have a different meaning: zero, via Arabic. In some languages it means digit.
- pinkmuffinere 10mo ago
  lol I never made that connection — in Turkish, zero is sıfır, which does sound a lot like cipher. Also, password is şifre, which again sounds similar. Looking online, apparently the path is sifr (Arabic, meaning zero) -> cifre (French, first meaning zero, then any numeral, then coded message) -> şifre (Turkish, code/cipher)
  - ls-a 10mo ago
    [flagged]
    - drivebyhooting 10mo ago
      This doesn’t sound right. What about Chinese?
      - jjtheblunt 10mo ago
        i'm quite sure the person was joking
      - astrange 10mo ago
        Basque and Pirahā are the good ones.
    - rmunn 10mo ago
      > All world languages are a deviation from Arabic Spouse of a linguist here. That is absolutely not true. To summarize a LOT, there are multiple languages that share common roots, which linguists classify into language "families". If you go to https://en.wikipedia.org/wiki/List_of_language_families#Spoken_language_families https://en.wikipedia.org/wiki/List_of_language_families#Spok... and sort the list by number of current speakers (which adds up to far more than the population of the world because so many people speak two or more languages), you'll find the top five language families are Indo-European (which includes most European languages, including English), Sino-Tibetan (which includes Chinese), Atlantic-Congo (which includes Bantu and many other languages spoken in Africa, most of which you probably won't have heard of unless you're a linguist or you live in Africa), Afroasiatic (which includes Arabic), and Austronesian (which includes Tagalog, which you might know by the name Filipino). It might be possible to claim that the Afroasiatic languages are all derived from Arabic, but the only influence that the Arabic language has had on Indo-European languages such as English is via loanwords (like algebra, for example). This does not make English a derivative of Arabic any more than Japanese (which has borrowed several English words such as カメラ, "kamera", from camera) is a derivative of English. Borrowing a word, or even a few dozen words, from another language does not make it a derivative. English, while it gleefully borrows loanwords from everywhere, is derived from French and German (or, to be more accurate, from Anglo-Norman and Proto-Germanic).
kazinator 10mo ago
> Dodgson’s original paper from 1867 is quite readable, surprisingly so given that math notation and terminology changes over time. Given that Jabberwocky is also quite readable, we shouldn't be too astonished.
- saghm 10mo ago
  I think in this case "readable" means "comprehensive", which maybe doesn't apply quite as much to Jabberwocky (albeit by design).
- JadeNB 10mo ago
  > Given that Jabberwocky is also quite readable, we shouldn't be too astonished. The conventions of literature have changed a lot less than math notation and terminology have since 1867.
01jonny01 10mo ago
When I'm not cognitively depleted from over working and kids I'd really like to sit down and read this properly.
howling 10mo ago
Terrence Tao blogged about this. https://terrytao.wordpress.com/2017/08/28/dodgson-condensation-from-schur-complementation/ https://terrytao.wordpress.com/2017/08/28/dodgson-condensati...
BobbyTables2 10mo ago
Wow, I never realized the cofactor method wasn’t the only one. I loathed it and it put me off wanting to get into advanced matrix topics.
- RossBencina 10mo ago
  I don't think determinants play a central role in modern advanced matrix topics. Luckily for me I read Axler's "Linear Algebra Done Right" (which uses determinant-free proofs) during my first linear algebra course, and didn't concern myself with determinants for a very long time. Edit: Beyond cofactor expansion everyone should know of at least one quick method to write down determinants of 3x3 matrices. There is a nice survey in this paper: Dardan Hajriza, "New Method to Compute the Determinant of a 3x3 Matrix," International Journal of Algebra, Vol. 3, 2009, no. 5, 211 - 219. https://www.m-hikari.com/ija/ija-password-2009/ija-password5-8-2009/hajrizajIJA5-8-2009.pdf https://www.m-hikari.com/ija/ija-password-2009/ija-password5...
  - srean 10mo ago
    > I don't think determinants play a central role in modern advanced matrix topics. Not true at all. It's integral to determinantal stochastic point processes, commute distances in graphs, conductance in resistor networks, computing correlation via linear response theory, enumerating subgraphs, representation theory of groups, spectral graph theory... I am sure many more
  - omnicognate 10mo ago
    The 4th edition of Linear Algebra Done Right has a much improved approach to determinants themselves (still relegated to the end, where it should be). From the list of improvements: > New Chapter 9 on multilinear algebra, including bilinear forms, quadratic forms, multilinear forms, and tensor products. Determinants now are defned using a basis-free approach via alternating multilinear forms. The basis-free definition is really rather lovely.
barbazoo 10mo ago
And just like back in university I know how how calculate Determinants but have no clue what one would actually use it for.
- Razengan 10mo ago
  3blue1brown is your friend
- ComplexSystems 10mo ago
  Suppose you have (let's say) a 3x3 matrix. This is a linear transformation that maps real vectors to real vectors. Now let's say you have a cube as input with volume 1, and you send it into this transformation. The absolute value of the determinant of the matrix tells you what volume the transformed cube will be. The sign tells you if there is a parity reversal or not.
- RossBencina 10mo ago
  Form a quadratic equation to solve for the eigenvalues x of a 2x2 matrix (|A - xI| = 0). The inverse of a matrix can be calculated as the classical adjugate multiplied by the reciprocal of the determinant. Use Cramer's Rule to solve a system of linear equations by computing determinants. Reason that if x is an eigenvalue of A then A - xI has a non-trivial nullspace (using the mnemonic |A - xI| = 0).
- adrian_b 10mo ago
  As another poster has also said, the determinant of a matrix provides 2 very important pieces of information about the associated linear transformation of the space. The sign of the determinant tells you whether the linear transformation includes a mirror reflection of the space, or not. The absolute value of the determinant tells you whether the linear transformation preserves the (multi-dimensional) volume (i.e. it is an isochoric transformation, which changes the shape without changing the volume), or it is an expansion of the space or a compression of the space, depending on whether the absolute value of the determinant is 1, greater than 1 or less than 1. To understand what a certain linear transformation does, one usually decomposes it in several kinds of simpler transformations (by some factorization of the matrix), i.e. rotations and reflections that preserve both size and shape (i.e. they are isometric transformations), isochoric transformations that preserve volume but not shape, and similitude transformations (with the scale factor computed from the absolute value of the determinant), which preserve shape, but not volume. The determinant provides 2 of these simpler partial transformations, the reflection and the similitude transformation.
gsf_emergency_6 10mo ago
http://www.gutenberg.org/files/37354/37354-pdf.pdf http://www.gutenberg.org/files/37354/37354-pdf.pdf

209 points by tzury 10mo ago

messe 10mo ago

HN title filter cut off the initial "How". You can manually edit it back in.

marcusestes 10mo ago
“Drop the ‘how.’ It’s cleaner.”
- deleted 10mo ago
  [deleted]
- vharuck 10mo ago
  It gives it a different implication. As I read it, an article titled "Lewis Carroll Computed Determinates" has three possible subjects: 1. Literally, Carroll would do matrix math. I know, like many on HN, that he was a mathematician. So this would be a dull and therefore unlikely subject. 2. Carroll invented determinates. This doesn't really fit the timeline of math history, so I doubt it. 3. Carroll computed determinates, and this was surprising. Maybe because we thought he was a bad mathematician, or the method had recently been invented and we don't know how he learned of it. This is slightly plausible. 4. (The actual subject). Carroll invented a method for computing determinates. A mathematician inventing a math technique makes sense, but the title doesn't. It'd be like saying "Newton and Leibnitz Used Calculus." Really burying the lede. Of course, this could've been avoided had the article not gone with a click-bait style title. A clearer one might've been "Lewis Carroll's Method for Calculating Determinates Is Probably How You First Learned to Do It." It's long, but I'm not a pithy writer. I'm sure somebody could do better.
  - SAI_Peregrinus 10mo ago
    "Lewis Carroll's Method for Computing Determinates" is headlinese and avoids the "how".
  - miltonlost 10mo ago
    "How Lewis Carroll Computed Determinates" is fine and not clickbait because it provides all the pertinent information and is an accurate summary of its contents. Clickbait would be "you would never guess how this author/mathematician computed determinants" since it requires a clickthrough to know who the person is. How is perfectly fine IMO to have in the title because I personally would expect the How to be long enough to warrant a necessary clickthrough due to the otherwise required title length.
    - 867-5309 10mo ago
      it's not quite McKean's Law so I'll settle for contagious

esafak 10mo ago

> Arrange the given block, if necessary, so that no ciphers [zeros] occur in its interior. I forgot that cipher used to have a different meaning: zero, via Arabic. In some languages it means digit.

pinkmuffinere 10mo ago
lol I never made that connection — in Turkish, zero is sıfır, which does sound a lot like cipher. Also, password is şifre, which again sounds similar. Looking online, apparently the path is sifr (Arabic, meaning zero) -> cifre (French, first meaning zero, then any numeral, then coded message) -> şifre (Turkish, code/cipher)
- ls-a 10mo ago
  [flagged]
  - drivebyhooting 10mo ago
    This doesn’t sound right. What about Chinese?
    - jjtheblunt 10mo ago
      i'm quite sure the person was joking
    - astrange 10mo ago
      Basque and Pirahā are the good ones.
  - rmunn 10mo ago
    > All world languages are a deviation from Arabic Spouse of a linguist here. That is absolutely not true. To summarize a LOT, there are multiple languages that share common roots, which linguists classify into language "families". If you go to https://en.wikipedia.org/wiki/List_of_language_families#Spoken_language_families https://en.wikipedia.org/wiki/List_of_language_families#Spok... and sort the list by number of current speakers (which adds up to far more than the population of the world because so many people speak two or more languages), you'll find the top five language families are Indo-European (which includes most European languages, including English), Sino-Tibetan (which includes Chinese), Atlantic-Congo (which includes Bantu and many other languages spoken in Africa, most of which you probably won't have heard of unless you're a linguist or you live in Africa), Afroasiatic (which includes Arabic), and Austronesian (which includes Tagalog, which you might know by the name Filipino). It might be possible to claim that the Afroasiatic languages are all derived from Arabic, but the only influence that the Arabic language has had on Indo-European languages such as English is via loanwords (like algebra, for example). This does not make English a derivative of Arabic any more than Japanese (which has borrowed several English words such as カメラ, "kamera", from camera) is a derivative of English. Borrowing a word, or even a few dozen words, from another language does not make it a derivative. English, while it gleefully borrows loanwords from everywhere, is derived from French and German (or, to be more accurate, from Anglo-Norman and Proto-Germanic).

kazinator 10mo ago

> Dodgson’s original paper from 1867 is quite readable, surprisingly so given that math notation and terminology changes over time. Given that Jabberwocky is also quite readable, we shouldn't be too astonished.

saghm 10mo ago
I think in this case "readable" means "comprehensive", which maybe doesn't apply quite as much to Jabberwocky (albeit by design).
JadeNB 10mo ago
> Given that Jabberwocky is also quite readable, we shouldn't be too astonished. The conventions of literature have changed a lot less than math notation and terminology have since 1867.

01jonny01 10mo ago

When I'm not cognitively depleted from over working and kids I'd really like to sit down and read this properly.

howling 10mo ago

Terrence Tao blogged about this. https://terrytao.wordpress.com/2017/08/28/dodgson-condensation-from-schur-complementation/ https://terrytao.wordpress.com/2017/08/28/dodgson-condensati...

BobbyTables2 10mo ago

Wow, I never realized the cofactor method wasn’t the only one. I loathed it and it put me off wanting to get into advanced matrix topics.

RossBencina 10mo ago
I don't think determinants play a central role in modern advanced matrix topics. Luckily for me I read Axler's "Linear Algebra Done Right" (which uses determinant-free proofs) during my first linear algebra course, and didn't concern myself with determinants for a very long time. Edit: Beyond cofactor expansion everyone should know of at least one quick method to write down determinants of 3x3 matrices. There is a nice survey in this paper: Dardan Hajriza, "New Method to Compute the Determinant of a 3x3 Matrix," International Journal of Algebra, Vol. 3, 2009, no. 5, 211 - 219. https://www.m-hikari.com/ija/ija-password-2009/ija-password5-8-2009/hajrizajIJA5-8-2009.pdf https://www.m-hikari.com/ija/ija-password-2009/ija-password5...
- srean 10mo ago
  > I don't think determinants play a central role in modern advanced matrix topics. Not true at all. It's integral to determinantal stochastic point processes, commute distances in graphs, conductance in resistor networks, computing correlation via linear response theory, enumerating subgraphs, representation theory of groups, spectral graph theory... I am sure many more
- omnicognate 10mo ago
  The 4th edition of Linear Algebra Done Right has a much improved approach to determinants themselves (still relegated to the end, where it should be). From the list of improvements: > New Chapter 9 on multilinear algebra, including bilinear forms, quadratic forms, multilinear forms, and tensor products. Determinants now are defned using a basis-free approach via alternating multilinear forms. The basis-free definition is really rather lovely.

barbazoo 10mo ago

And just like back in university I know how how calculate Determinants but have no clue what one would actually use it for.

Razengan 10mo ago
3blue1brown is your friend
ComplexSystems 10mo ago
Suppose you have (let's say) a 3x3 matrix. This is a linear transformation that maps real vectors to real vectors. Now let's say you have a cube as input with volume 1, and you send it into this transformation. The absolute value of the determinant of the matrix tells you what volume the transformed cube will be. The sign tells you if there is a parity reversal or not.
RossBencina 10mo ago
Form a quadratic equation to solve for the eigenvalues x of a 2x2 matrix (|A - xI| = 0). The inverse of a matrix can be calculated as the classical adjugate multiplied by the reciprocal of the determinant. Use Cramer's Rule to solve a system of linear equations by computing determinants. Reason that if x is an eigenvalue of A then A - xI has a non-trivial nullspace (using the mnemonic |A - xI| = 0).
adrian_b 10mo ago
As another poster has also said, the determinant of a matrix provides 2 very important pieces of information about the associated linear transformation of the space. The sign of the determinant tells you whether the linear transformation includes a mirror reflection of the space, or not. The absolute value of the determinant tells you whether the linear transformation preserves the (multi-dimensional) volume (i.e. it is an isochoric transformation, which changes the shape without changing the volume), or it is an expansion of the space or a compression of the space, depending on whether the absolute value of the determinant is 1, greater than 1 or less than 1. To understand what a certain linear transformation does, one usually decomposes it in several kinds of simpler transformations (by some factorization of the matrix), i.e. rotations and reflections that preserve both size and shape (i.e. they are isometric transformations), isochoric transformations that preserve volume but not shape, and similitude transformations (with the scale factor computed from the absolute value of the determinant), which preserve shape, but not volume. The determinant provides 2 of these simpler partial transformations, the reflection and the similitude transformation.

gsf_emergency_6 10mo ago

http://www.gutenberg.org/files/37354/37354-pdf.pdf http://www.gutenberg.org/files/37354/37354-pdf.pdf