Fun fact: zero and numerals were not invented by the Arabs. The Arabs learnt the concept & use of mathematical zero, numerals, decimal system, mathematical calculations, etc. from the ancient Hindus/Indians.
And from the Arabs, the Europeans learnt it.
Persian scholar Al Khwarizmi translated and used the Hindu/Indian numerals (including concept of mathematical zero) and "Sulba Sutras" (Hindu/Indian methods of mathematical problem solving) into the text Al-Jabr, which the Europeans translated as "Algebra" (yup, that branch of mathematics that all schoolkids worldwide learn from kindergarten).
Yeah I believe modern trigonometry and the terms sine and cos also trace their origins to Sanskrit through Arabic. It's a shame that ancient/medieval India contributed so much to science and math but hasn't been able to innovate in centuries past :(
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)
Nice! Imagine the second meaning going back to Arabic and now it's a full loop! It can even override the original meaning given enough time and popularity (not especially for "zero", but possibly for another full-loop word).
The Turkish password word may be the same used for signature? I suspect so, because in Greek we have the Greek word for signature but also a Turkish loan word τζίφρα (djifra).
When someone says "it still means zero" about Tamil when responding to comments about Arabic, two languages which have no shared root and little similarity, what does that mean?
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:
> 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
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.
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.
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.
Here are three reasons you want to be able to calculate the volume change for arbitrary parallelpipeds:
- If det M = 0, then M is not invertible. Knowing this is useful for all kinds of reasons. It means you cannot solve an equation like Mx = b by taking the inverse ("dividing") on both sides, x = M \ b. It means you can find the eigenvalues of a matrix by rearranging Mx = λx <--> (M-λI)x = 0 <--> det M-λI = 0, which is a polynomial equation.
- Rotations are volume-preserving, so the rotation group can be expressed as the matrices where det M = 1 (well, the component connected to the identity). This is useful for theoretical physics, where they're playing around with such groups and need representations they can do things with.
- In information theory, the differential entropy (or average amount of bits it takes to describe a particular point in a continuous probability distribution) increases if you spread out the distribution, and decreases if you squeeze it together by exactly log |det M| for a linear transformation. A nonlinear transformation can be linearized with its gradient. This is useful for image compression (and thus generation) with normalizing flow neural networks.
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).
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.
> 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.
Fun fact: zero and numerals were not invented by the Arabs. The Arabs learnt the concept & use of mathematical zero, numerals, decimal system, mathematical calculations, etc. from the ancient Hindus/Indians. And from the Arabs, the Europeans learnt it.
https://en.wikipedia.org/wiki/Hindu-Arabic_numeral_system
Persian scholar Al Khwarizmi translated and used the Hindu/Indian numerals (including concept of mathematical zero) and "Sulba Sutras" (Hindu/Indian methods of mathematical problem solving) into the text Al-Jabr, which the Europeans translated as "Algebra" (yup, that branch of mathematics that all schoolkids worldwide learn from kindergarten).
The word used to mean "empty" (and not algebraic zero) in both Arabic and Sanskrit.
https://www.open.ac.uk/blogs/MathEd/index.php/2022/08/25/the...
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Yeah I believe modern trigonometry and the terms sine and cos also trace their origins to Sanskrit through Arabic. It's a shame that ancient/medieval India contributed so much to science and math but hasn't been able to innovate in centuries past :(
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And the English word 'algorithm' comes from Al Khwarizmi's name[1].
1: https://en.wikipedia.org/wiki/Al-Khwarizmi#:~:text=His%20nam...
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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)
Nice! Imagine the second meaning going back to Arabic and now it's a full loop! It can even override the original meaning given enough time and popularity (not especially for "zero", but possibly for another full-loop word).
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The Turkish password word may be the same used for signature? I suspect so, because in Greek we have the Greek word for signature but also a Turkish loan word τζίφρα (djifra).
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In Romanian:
- cifru -> cipher
- cifră -> digit
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[flagged]
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In Tamil, it still means a zero. It's usually pronounced like 'cyber' though, because Tamil doesn't have the 'f'/'ph' sound natively.
When someone says "it still means zero" about Tamil when responding to comments about Arabic, two languages which have no shared root and little similarity, what does that mean?
I think it means HN is full of misleading ideas.
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Dutch too: "Cijfer", German, "Ziffer", French: "Chifre", Spanish: "Cifra".
Swedish: "Siffra"
That explains Major Zero and his organisation Cipher in the metal gear series
Terrence Tao blogged about this.
https://terrytao.wordpress.com/2017/08/28/dodgson-condensati...
> 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.
> 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.
I think in this case "readable" means "comprehensive", which maybe doesn't apply quite as much to Jabberwocky (albeit by design).
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.
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...
> 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
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.
When I'm not cognitively depleted from over working and kids I'd really like to sit down and read this properly.
http://www.gutenberg.org/files/37354/37354-pdf.pdf
And just like back in university I know how how calculate Determinants but have no clue what one would actually use it for.
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.
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.
Here are three reasons you want to be able to calculate the volume change for arbitrary parallelpipeds:
- If det M = 0, then M is not invertible. Knowing this is useful for all kinds of reasons. It means you cannot solve an equation like Mx = b by taking the inverse ("dividing") on both sides, x = M \ b. It means you can find the eigenvalues of a matrix by rearranging Mx = λx <--> (M-λI)x = 0 <--> det M-λI = 0, which is a polynomial equation.
- Rotations are volume-preserving, so the rotation group can be expressed as the matrices where det M = 1 (well, the component connected to the identity). This is useful for theoretical physics, where they're playing around with such groups and need representations they can do things with.
- In information theory, the differential entropy (or average amount of bits it takes to describe a particular point in a continuous probability distribution) increases if you spread out the distribution, and decreases if you squeeze it together by exactly log |det M| for a linear transformation. A nonlinear transformation can be linearized with its gradient. This is useful for image compression (and thus generation) with normalizing flow neural networks.
Rotations have determinant 1, but not all matrices of determinant 1 in the connected component of the identity are rotations
3blue1brown is your friend
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).
HN title filter cut off the initial "How".
You can manually edit it back in.
“Drop the ‘how.’ It’s cleaner.”
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.
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