I think if anything makes it “scary“ it's just that Who would expect a prime number, out of the blue like 17, to appear in an analysis of a recreational, vocally spawned series, and appear under a sqrt radical yet? And if you want something scarier see the cool RUclips videos on finding the sqrt of i, and the one deriving (i ^ i) which turns out to be a REAL number! (I double-checked the latter, [ i ^ i ], with my TI-89, and it confirmed it.)
Fun fact about Eigenvalues and Eigenvectors: They were invented by a Dutch mathematician, who decided to name them "eigenwaarden (Eigenvalues)" en "eigenvectoren (Eigenvectors)". Literally translated they mean "[The matrix'] own values" and "[The matrix'] own vectors". However, when they were shown to some English mathematicians, something went wrong in translation, and the English thought that they were invented by some German mathematician called Eigen, which is why the terms are capitalized.
@@NorbiPeti In Finnish, they’re called: _”Ominaisarvot”_ and: _”Ominaisvektorit”_ (literally: ”Characteristic Values” and: ”Characteristic Vectors”), and we also have something called: _”Ominaisavaruus”_ (”Characteristic Space”; a collection of all the Eigenvectors with the same Eigenvalue). 🇫🇮🇭🇺
I was wondering why you looked at the binary version rather than the classic base-10, so I looked it up. Turns out the ratio of successive terms in the base-10 sequence converges to λ = 1.303577... where λ is an algebraic number with a minimal polynomial of degree 71; i.e. λ is a root of a polynomial of degree 71 and no smaller polynomial, a fact which was proven by Conway.
@@iveharzing Yeah, λ is the unique positive real root of the polynomial x⁷¹ − x⁶⁹ − 2x⁶⁸ + . . . + 3x − 6. The full polynomial is printed here: en.wikipedia.org/wiki/Look-and-say_sequence#Conway's_constant_as_a_polynomial_root.
And the same politinomial works for base 89 ;) (Or any base bigger than 3, since the look and say sequence will never have 4 consecutive equal digits in bases that have a symbol for 3).
Yeah, and if you know a lot of different kinds of mathematical sequences, it probably just means you'll waste more time checking if any of them work, before it dawns on you to just say the numbers out loud.
the mathematician's answer to those "puzzles" is always the same: There is no unique solution. There are infinitely many sequences starting with those numbers. Some are finite, some are infinite. A simple one is always: "They are the zeros of some polynomial, which I can work out if you want me to."
@@sebastianjost True, but we're not in a vacuum. When someone gives you a sequence and asks for the pattern, the conventional assumption is that the pattern is clever and unique. A pattern that can be generated for any given sequence is neither unique nor clever
I always get a slightly bitter taste whenever I think about it because I remember getting it wrong, and it feels like having met an old friend or a celebrity, making a terrible impression on them, and having them leave with that image of you, it always feels a little sad how I'll never get another first time trying to guess what it is haha
I think conjectures are like politicians, they have to look good but not so good that you start to wonder if they spend more time on their appearance than their actual political work :)
I've spend over a decade looking at eigenvalues/vectors on an almost daily basis (for exactly the reason Matt gives at 10:10), to the point that I'd completely forgotten it was something most people have never heard of. First, I felt smug. Then, I felt nostalgia for simpler times. Now, I just feel bitterness and jealously towards people who haven't had the Pauli matrices and Cayley-Hamilton Theorem burnt so deep in their brains that they've literally come up in their dreams.
Knowledge is power, but the power comes with responsibility. You should be proud of yourself for bearing the responsibility that most people are too weak to do!😁
Brain: "Hey look at this. It's a 5x5 matrix. What's it's eigenvector?" Also Brain: "Which number is flurb? Also why is the matrix now a cube? Also why am I doing this while going down a waterslide that also happens to be the drainspout outside of my window?"
@@ANunes06 Is this how you have math in your dreams? How horrifying. In mine I just see nonsense mathematics and think "Hm. Interesting. I should study this!"
I'm at a place in my math learning where I was just able to anticipate the next step being eigenvectors, but nothing about how to apply them, so I did appreciate the crash course on them. I'm sure I'll need to learn them six or eleven more times before they really stick
@@whatzause since you're remarking on it, I suppose it's what feels like a comfy numerical rendering of "more than one or two handfuls of times", 5n + 1, because a hand has five fingers and all But dissecting it takes a little bit of the whimsy out of it, possibly
@@collin4555 Right. My comment was satirical of course. BTW, I had seen this series decades ago, but am surprised at its being discovered to have mathematical meaning. As ever, Matt’s presentations are super!
For what it's worth, this is an area that in my experience pretty much every professor re-teaches at the beginning of every course involving it. It's complicated and difficult to get a good intuition for, so yeah you will have to learn and forget it a bunch of times before it sticks and that's ok 😁 (Ofc each time the reminder builds on top of new knowledge from each part before but people myself included regularly lose the basics of eigenvectors each time we apply it in new and amazing ways, I must have watched the mentioned excellent threeblue1brown video at least 10 times in the last 5 or so years )
Feeling a bit proud figuring out the first sequence in ~60 seconds when Matt mentioned it taking him all day. The thing that got my attention was the three consecutive 1's just above the only given '3'. That was enough to give away the trick!
I'm a materials science student, so I've had to deal with Eigenvalues and Eigenvectors a lot. Needless to say, thanks for the segment explaining them! Because I've completely forgotten what they are and how they work-
I learnt about eigenvalues/vectors in a 2nd year maths unit as part of my eng degree. Sadly, I have never used them again since. That unit was taught poorly, most of us could barely understand what the lecturer was saying. Your explanation was much better, and an interesting application.
I finally used eigenvalues and eigenvectors in my engineering masters degree, and in two completely different situations. The solution to problems in structural dynamics is effectively an eigenvalue problem. And they also come up when you consider stresses and strains on a 3D element. So just within structural engineering there are at least two ways eigenvalues/vectors are useful. No doubt there are countless applications in other fields.
I had the same experience! For some reason the class on matrices was taught over the summer when most of the school was closed. So all math majors were required to stay on alone and try to learn this stuff in the heat from a professor who was too old to still be teaching. He would make minor arithmetic mistakes in every matrix, and if we attempted to point it out, it would take 5 minutes for him to find what digit we were referring to and correct it. After a while, we wouldn't tell him, but just note it quietly between ourselves. Although I learned enough to pass, I never understood it.
I was so fking happy when you showed the flowchart and I immediately thought of Markow-chains. In highschool, we learned this rather extensively and I was so glad I was able to recognize the application of something I learned just before you revealed it. Idk, it makes me so happy. Edit: loled when you delivered that genius punchline.
Thank you for explaining the use of Eigenvectors and Eigenvalues. My entire maths course in uni never actually explained why we might actually want to find them!
These are very important for lots of applications in science and engineering. E. g. the eigenvectors of the moment of inertia tensor of a rigid body tells you the axis around which the body can have stable rotation, and the corresponding eigenvalues are the moments of inertia for these axes. In quantum mechanics, these are even more important. Essentially, every thing which you can measure corresponds to a matrix, and the only values which can be measured are the eigenvalues of that matrix.
Recognised that sequence instantly. I learned it around a decade ago when I first played Knights of the Old Republic. It was one of the many puzzles they had in that game and I guess it stuck with me since I still remember it to this day.
Over the course of a couple of minutes you made me understand eigenvalues and eiganvectors better than an entire unit on the topic in Linear Algebra class back in college.
My high school LinAlg class didn't even cover eigenvalues and eigenvectors. We did other stuff that was questionable in terms of whether it should belong in that class.
@@vigilantcosmicpenguin8721 Technically your comment raises the number of examples for this yet to be proven conjecture to at least 2, assuming you are not the same bored mathematician who inspired the video.
When I saw eigenvectors and eigenvalues flash up on screen it felt good remembering what those were from the little bit of linear algebra I did during college. I had a great professor. I kinda wish I had been going deeper into mathematics itself and not eventually shifted gears to computer science. He told us if we took his grad courses he would go into depth about the proof that got him his doctorate. Maybe I'll go back one day before he retires. I'm sure I could look it up but it hits differently when the person who discovered something explains it to you.
There was a question about eigenvalues on university challenge the other day, and I am ashamed to admit that I could only remember the term eigenvectors and that I had studied them at some point!
You can still learn some deeper mathematics. A good side project is writing an algebraic geometry program, just the simple stuff like declaring polynomial rings, irreducibility criterion, ideals, Grobner bases, stuff like that. It’s related to computer science and maths
Would love to see you do a video on the chess cheating controversy regarding Hans Niemann. The probabilities regarding this controversy are highlighted in Hikaru’s reaction video to the one Yosha posted. It might take a while to compile the data you would want, but it seems like something you would be interested in.
@@timseguine2 Ich dachte, die englischsprachige Mathematikergesellschaft hat "Ansatz" bereits anglisiert, so wie es Matt mit "Eigen" (I can) bereits gemacht hat.
I stared at the puzzle at the start of the video for like 10 seconds before cracking it, and that euphoric high that came with the realization that I solved a puzzle in seconds, that an expert spent an entire day on cracking, is something I don't think is conceptually describable.
For those of you playing educational RUclipsr bingo, Matt mentioned the names of Crash Course and the English translation of Kurzgesagt (In a Nutshell), as well as 3Blue1Brown. Well done Mr. Parker, well done, even if the first two were simply expressions you used and not intentional, which, let’s be honest, they likely were.
Matt: "A good reminder, just because it looks like something is the answer in mathematics, we don't know for certain until we do the maths" I shall Burn this into mind, it is one of the many things I struggle with, I'm constantly trying to eye-ball things. Amazing video.
For me, the most hilarious thing of all Matt's videos is to see how he primarily amuses himself with those Daddy maths jokes :D It totally cracks me every time :D
@@WreckedRectum Yeah, one of the better ones on rewatching was when he said "I hope you can, uh... matri-see? how eigenvalues and eigenvectors work". He kept a straight face for a split second before he just had to grin, and then he was so proud of the dad joke that he very slowly blinked. That slow blink was such a giveaway. I loved it!
I do remember seeing these kind of questions in maths books, and I learned to hate the very sight of them. Because they would appear, a bunch of stings of random numbers. And with no hint of any solution it just says "find the next number in the sequence". And in the answers it would be just a string of equally random seeming numbers. No hint of how one should arrive at that particular one. What the criteria for success was or anything. The result was that those who decrypted these hieroglyphic things felt smart and everyone else, me included, felt stupid. And encountering enough of similar tasks with no guidance, and no pedagogical help to be found. I just lumped them together as "one of those". I never got the tools to crack them, I was just left there, abandoned. Just writing a random number because a fail at least made it go away. This, and the fact that the maths books in general I encountered were seemingly written backwards. Making me do tasks before teaching me how to do them. Making me wonder if I was having a stroke, if I was having early onset dementia, then I turn the page and it explained the mechanics to solve the past few pages tasks. Sorry about the rant, I just had flashbacks to how exclusionary maths can be at times. I think it's part of why I liked computer classes, they at least knew how to teach. And in general, I knew what a success state was. I like finding out about math stuff nowadays. 15-20 years after those schoolbooks. I just wonder how many mathematical minds were lost, because of horribly written books.
It's interesting to look at the binary look-and-say sequences with other starting points. The sequence starting with 0 → 10 → 1110 → ... looks a bit different (all end with 0, of course). You'll still get the same block graph, just without 1 and 11 [which are irrelevant anyways], so we'll expect to get the same ratio of ones and zeroes in the limit. But not every starting point gives such a sequence, for example when starting with 111, we just stay at 111, so the limit would be 0 (or ∞, depending on the point of view). Are there other sequences which don't grow forever?
@@cmcaulay07 Not only can you "encode the digits with representations higher than the base", he shows it being done at 5:04 with 1110->11110 and 11110->100110
@@cmcaulay07 > "You can't encode the digits with representations higher than the base" > binary > "you have to either pick "one one, two ones (11101), or two ones, one one (10111)"." I lol'd.
I doubt there's any other sequences that don't grow forever. Conway's solution for the original only works because of a theorem that every sequence eventually decays into a sequence of the atoms, and from that it's easy to see that the sequence 22 is the only one that could possibly work. (Everything else decays into something longer (after a step or two), and nothing else decays into specifically 22.) There's no such supporting theorem here - after all, 111 literally isn't an atom in this structure. But I still don't see anything else as possibly being able to work, and I'd even expect adding 111 as an atom to make an equivalent theorem to Conway's to hold.
It's interesting how much this video resonates with Sabine Hossenfelder's "Lost in Math", which is very much worth reading on the subject of elegance as a goal in math and physics.
This game doesn't really work in spanish, since the word for the number 1 is different than the word for "1 of something" ("uno" and "un"). So if you say "un uno" (kind of like "a 1") it's different that saying "uno uno" ("one one").
@@omp199 yeah, but I guess the algorithm is still coherent beyond language: every time you start a new row of numbers state two inputs : 1st how many 2nd of which number, then keep going..
As soon as I saw the graph I knew I'd be "blessed" to see matrices pop up. I remember having to deal with a similar problem when I was prototyping a game and ran into a similar problem relating to random walks which turned to markov chains, which turned to eigenvalue shenanigans. Cool math, but not exactly transparent to the people playing the game or immediately intuitive.
I love it when you give a number sequence and people think there's some sort of advanced algebraic or geometric relationship then its just... nope... Numbers.
I did a lot of engineering courses, saw eigenvalues and eigenvectors until I could claim I know them by heart. I thought that that segment wouldn't teach me anything new, but it did. I finally made the mental connection on why we actually use them for solving systems of differential equations! Thank you, Matt!
Can you do a video on Hans Niemann allegedly cheating in the chess Sinqued cup? I would love a statistical breakdown the same way you did with Dreams mc world record
There's not a unique eigenvector corresponding to an eigenvalue. You can scale an eigenvector by a constant and it's still an eigenvector for the same eigenvalue. In fact, you can take any vector from the eigenvalue's eigenspace.
Very interesting video! A small note: your audio is quite quiet, I had to max my speakers to reach a "normal" volume. Maybe 2x as loud would be perfect? Appreciate you!
That's so crazy. I was reading about quantum entanglement the other day and stumbled upon eigenvalues/ -vectors as they relate to the wave function collapse. "Everything is, like, connected, man."
I literally just finished studying eigen vectors and values like, a week ago (although it wasn’t called like that in my course)! What a coincidence! I wonder if Maple uses the same algorithm for finding these values as we used, one of the task was to write a program that finds them. There is a way to find two pairs of answers, to be specific, to derive one pair from another, which is really neat!
@0:19 It's a read number. starting at 1. 1; one 1; two 1's; one 2, one 1; one 1, one 2, two 1's; three 1's, two 2's, one 1. Next: one 3, one 1, two 2's, two 2's. I'll continue watching in case there's an alternate sequence, but this one fits so far. The word one looks so weird right now. @1:00 didn't know there was a name for it. I've seen enough to make the comment.
Hey, I did eigenvectors and eigenvalues in economics! We did it for one module, for one exam, and I've completely forgetten everything about them ever since.
I think it's amazing that someone was able to completely solve this problem, but we then have problems like the Collatz conjecture or the Lychrel number problem that can't be solved.
Look and say we're seeing saying13112221. When you take this sequence out to Infinity you still only ever see the integers 1 2 and 3 no other integers appear in the sequence. Also if you create a sequence that counts the amount of digits in each of the numbers of the look and say sequence and then create a ratio of two consecutive numbers in this new sequence as it approaches Infinity it equals a constant that is the unique positive real root of a polynomial of degree 71. The constant is called Conway's constant and it is approximately 1.303577...
I felt like I was having a stroke during this explanation. I'm too old to have learned matrices in school, so this whole video felt like a definition of terms using only the terms defined within the definition. "The animal a dog is an animal that is a dog, and excludes animals that are not dogs. Even animals that are similar to dogs, but not dogs, are animals that are not dogs. Only dogs are dogs. Here is a proof."
To calculate the probability of an arbitrary elegant conjecture being true, we need to find the set of all elegant conjectures, and divide it into True, False, Unknown, and Not Well Defined. A strict measure of Elegance will likely involve how many words or ideas are needed to describe the conjecture. If we pick a maximum number of words at say 20, and look at all (words)^20 items, we must note the are more conjectures than can be examined in a reasonable amount of time. Thus the conjecture 'elegant conjectures are trust-able' is in the Unknown category, for at least the age of one universe.
9:21 shouldn't the resulting vector be [1,-1] with the eigenvalue being -1? At the moment that's flipping the sign repeatedly rather than showing the actual eigenvalue
What I find more interesting about this sequence (the base 10 one) is the question: "can a 4 or larger digit appear in the sequence?". It is easy to get the number 3, for example ...12... -> ...1112... -> ...31... In the first number I assume that it is proceeded by non-1 digit and followed by a non-2 digit. The second number must be proceeded by the same non-1 digit as the first number, but since we can not be sure how many non-2 digits were in the first number, we do not know whether the following digit is or is not a 2. This means that in the third number all we know for sure is that there are 3 1's in a row, but it shows that it is possible for a 3 to exist in a number in the sequence. If you start the sequence with a digit larger than 3 it will be replicated in all subsequent numbers, but can you get one to spontaneously appear as a result of the sequence operation?
In order for a digit larger than 3 to spontaneously appear as a result of the sequence operation, the prior sequence must look something like (1) BBBB... or (2) ...ABBBB... where I use ... to denote "the rest" of the sequence leftwards or rightwards. Could (1) or (2) have been produced by a sequence operation? For (1) we would read it as B lots of B, followed by B lots of B. But that can't happen. That is the same as 2B lots of B. For (2) it may be read differently depending on parity. - A lots of B, followed by B lots of B, followed by B lots of ? - ? lots of A, followed by B lots of B, followed by B lots of B but in either case, we have a situation of X lots of B, followed by Y lots of B. Which is the same as (X+Y) lots of B, which is impossible. Thus a digit larger than 3 can't spontaneously appear.
As far as I can tell : The only way to have the string "1111" appear in a result (so that 41 would appear in the following result), is to say "one 1, one 1". You will never say this; on a previous entry, you would have said "three 1s" or "two 1s". Similarly, you will also never say "two 2s, two 2s" or "two 3s, two 3s". I don't know how to prove this.
That matrix *is* the graph. Lots of graph algorithms have very pleasant formulations in terms of operations on their adjacency matrices, at least when you can fit the adjacency matrix in memory...
Oh man I feel like I figured that out by chance, or maybe it's my mind associating 11 with one-ty one, but I got a huge Smile when I figured the pattern out!
Spoilers for the intro! I have paused the video before any explanation was given, so this is just my best guess. - - - - - each term in the sequence is a string representation of the term before it. 312211 aka “three 1s, two 2s, one 1” describes the previous string “111221,” which describes “1211” - “one 1, one 2, two 1s.” Thus, the next term in the sequence, following 312211, is “13112221”
Brilliant video. I especially liked 5:12 where my physics brain was screaming IT‘S GONNA BE A MARKOV CHAIN. And it was. What an amazing link between such seemingly different problems! For anyone interested: go look up "stochastic matrix"
Thanks for reminding me about eigenvalues and eigenvectors. Haven't had to think about those maths since college, which was extremely novel to see appear in one of your videos.
I think Matt is the only person who can say "you get square roots of 17 everywhere" and get a laugh. I did laugh out loud.
It's like, oh, tell me about it! I hate those square roots of 17.
What’s so scary about sqrt(17)? I can estimate that in my head as 4.12-ish. It’s the square roots of really small numbers that I’m scared of!
 I think if anything makes it “scary“ it's just that Who would expect a prime number, out of the blue like 17, to appear in an analysis of a recreational, vocally spawned series, and appear under a sqrt radical yet? And if you want something scarier see the cool RUclips videos on finding the sqrt of i, and the one deriving (i ^ i) which turns out to be a REAL number! (I double-checked the latter, [ i ^ i ], with my TI-89, and it confirmed it.)
Me, a HCSSiM alum:
I see this as an absolute win!
SAME!
"Eigen See Clearly Now" cracked me up! Haha, I really love this channel; keep it up, Matt!
Fun fact about Eigenvalues and Eigenvectors: They were invented by a Dutch mathematician, who decided to name them "eigenwaarden (Eigenvalues)" en "eigenvectoren (Eigenvectors)". Literally translated they mean "[The matrix'] own values" and "[The matrix'] own vectors". However, when they were shown to some English mathematicians, something went wrong in translation, and the English thought that they were invented by some German mathematician called Eigen, which is why the terms are capitalized.
I was wondering why they are named this way, since in Hungarian they are just called 'own vectors'.
@@NorbiPeti In Finnish, they’re called: _”Ominaisarvot”_ and: _”Ominaisvektorit”_
(literally: ”Characteristic Values” and: ”Characteristic Vectors”), and we also have something called: _”Ominaisavaruus”_ (”Characteristic Space”; a collection of all the Eigenvectors with the same Eigenvalue). 🇫🇮🇭🇺
Leonard Eigen, very famous German mathematician!
Although not in en-US. In the States, it's always lowercase except at the beginning of a sentence.
@Flavianus So, pretty much directly translatable. 🤔
I was wondering why you looked at the binary version rather than the classic base-10, so I looked it up. Turns out the ratio of successive terms in the base-10 sequence converges to λ = 1.303577... where λ is an algebraic number with a minimal polynomial of degree 71; i.e. λ is a root of a polynomial of degree 71 and no smaller polynomial, a fact which was proven by Conway.
I didn't understand anything but that sounds cool!
So a polynomial with as the highest term _a*x^71_ ?
@@iveharzing Yes.
@@iveharzing Yeah, λ is the unique positive real root of the polynomial x⁷¹ − x⁶⁹ − 2x⁶⁸ + . . . + 3x − 6. The full polynomial is printed here: en.wikipedia.org/wiki/Look-and-say_sequence#Conway's_constant_as_a_polynomial_root.
And the same politinomial works for base 89 ;)
(Or any base bigger than 3, since the look and say sequence will never have 4 consecutive equal digits in bases that have a symbol for 3).
That first sequence is one of those "so simple it's annoying" puzzles! Great video Matt as always.
Yeah, and if you know a lot of different kinds of mathematical sequences, it probably just means you'll waste more time checking if any of them work, before it dawns on you to just say the numbers out loud.
the mathematician's answer to those "puzzles" is always the same: There is no unique solution. There are infinitely many sequences starting with those numbers. Some are finite, some are infinite.
A simple one is always: "They are the zeros of some polynomial, which I can work out if you want me to."
@@sebastianjost True, but we're not in a vacuum. When someone gives you a sequence and asks for the pattern, the conventional assumption is that the pattern is clever and unique. A pattern that can be generated for any given sequence is neither unique nor clever
I always get a slightly bitter taste whenever I think about it because I remember getting it wrong, and it feels like having met an old friend or a celebrity, making a terrible impression on them, and having them leave with that image of you, it always feels a little sad how I'll never get another first time trying to guess what it is haha
It took me about 20 seconds to get it, does that make me smart or stupid?
I think conjectures are like politicians, they have to look good but not so good that you start to wonder if they spend more time on their appearance than their actual political work :)
nice conjecture you got there
They’re also similar in that if they look like Boris Johnson then you know you’ve done something VERY wrong
Who said that?
By your reasoning the Goldbach conjecture is like a politician.
@@Gapiedaan It's very elegant. Almost a little _too_ elegant, don't you think?
I've spend over a decade looking at eigenvalues/vectors on an almost daily basis (for exactly the reason Matt gives at 10:10), to the point that I'd completely forgotten it was something most people have never heard of. First, I felt smug. Then, I felt nostalgia for simpler times. Now, I just feel bitterness and jealously towards people who haven't had the Pauli matrices and Cayley-Hamilton Theorem burnt so deep in their brains that they've literally come up in their dreams.
Knowledge is power, but the power comes with responsibility.
You should be proud of yourself for bearing the responsibility that most people are too weak to do!😁
Brain: "Hey look at this. It's a 5x5 matrix. What's it's eigenvector?"
Also Brain: "Which number is flurb? Also why is the matrix now a cube? Also why am I doing this while going down a waterslide that also happens to be the drainspout outside of my window?"
@@ANunes06 Is this how you have math in your dreams? How horrifying. In mine I just see nonsense mathematics and think "Hm. Interesting. I should study this!"
@@kingp1n817 I guess anything can be an interesting accomplishment if you delude yourself into thinking it is
what kind of work involves these kindcof maths?
Eigen see clearly now... that's the kind of quality content I subscribed for
i am extremely proud of the fact that it took me less than a minute to figure out the sequence at the start of the video.
The real question: is Eigen See Clearly Now heading to Spotify?
I wonder if he had to revenue-split the video because of the tune.
I'm at a place in my math learning where I was just able to anticipate the next step being eigenvectors, but nothing about how to apply them, so I did appreciate the crash course on them. I'm sure I'll need to learn them six or eleven more times before they really stick
Six or eleven? Maybe 6.16611...😊
@@whatzause since you're remarking on it, I suppose it's what feels like a comfy numerical rendering of "more than one or two handfuls of times", 5n + 1, because a hand has five fingers and all
But dissecting it takes a little bit of the whimsy out of it, possibly
@@collin4555 Right. My comment was satirical of course. BTW, I had seen this series decades ago, but am surprised at its being discovered to have mathematical meaning. As ever, Matt’s presentations are super!
For what it's worth, this is an area that in my experience pretty much every professor re-teaches at the beginning of every course involving it. It's complicated and difficult to get a good intuition for, so yeah you will have to learn and forget it a bunch of times before it sticks and that's ok 😁
(Ofc each time the reminder builds on top of new knowledge from each part before but people myself included regularly lose the basics of eigenvectors each time we apply it in new and amazing ways, I must have watched the mentioned excellent threeblue1brown video at least 10 times in the last 5 or so years )
Feeling a bit proud figuring out the first sequence in ~60 seconds when Matt mentioned it taking him all day. The thing that got my attention was the three consecutive 1's just above the only given '3'. That was enough to give away the trick!
I'm a materials science student, so I've had to deal with Eigenvalues and Eigenvectors a lot.
Needless to say, thanks for the segment explaining them! Because I've completely forgotten what they are and how they work-
I learnt about eigenvalues/vectors in a 2nd year maths unit as part of my eng degree. Sadly, I have never used them again since. That unit was taught poorly, most of us could barely understand what the lecturer was saying. Your explanation was much better, and an interesting application.
I finally used eigenvalues and eigenvectors in my engineering masters degree, and in two completely different situations. The solution to problems in structural dynamics is effectively an eigenvalue problem. And they also come up when you consider stresses and strains on a 3D element. So just within structural engineering there are at least two ways eigenvalues/vectors are useful. No doubt there are countless applications in other fields.
I had the same experience! For some reason the class on matrices was taught over the summer when most of the school was closed. So all math majors were required to stay on alone and try to learn this stuff in the heat from a professor who was too old to still be teaching. He would make minor arithmetic mistakes in every matrix, and if we attempted to point it out, it would take 5 minutes for him to find what digit we were referring to and correct it. After a while, we wouldn't tell him, but just note it quietly between ourselves. Although I learned enough to pass, I never understood it.
I was so fking happy when you showed the flowchart and I immediately thought of Markow-chains. In highschool, we learned this rather extensively and I was so glad I was able to recognize the application of something I learned just before you revealed it. Idk, it makes me so happy.
Edit: loled when you delivered that genius punchline.
Thank you for explaining the use of Eigenvectors and Eigenvalues. My entire maths course in uni never actually explained why we might actually want to find them!
These are very important for lots of applications in science and engineering. E. g. the eigenvectors of the moment of inertia tensor of a rigid body tells you the axis around which the body can have stable rotation, and the corresponding eigenvalues are the moments of inertia for these axes.
In quantum mechanics, these are even more important. Essentially, every thing which you can measure corresponds to a matrix, and the only values which can be measured are the eigenvalues of that matrix.
6:38 perhaps the greatest moment of this channel so far
Recognised that sequence instantly. I learned it around a decade ago when I first played Knights of the Old Republic. It was one of the many puzzles they had in that game and I guess it stuck with me since I still remember it to this day.
Over the course of a couple of minutes you made me understand eigenvalues and eiganvectors better than an entire unit on the topic in Linear Algebra class back in college.
My high school LinAlg class didn't even cover eigenvalues and eigenvectors. We did other stuff that was questionable in terms of whether it should belong in that class.
Thank you for the simplest and clearest explanation of eigenvectors/eigenvalues I've ever seen!
This is proof of the theorem that mathematicians are bored.
Technically all it proves is ∃ mathematician : mathematician is bored. The theorem that ∀ mathematician : mathematician is bored is yet to be proven.
@@vigilantcosmicpenguin8721 Technically your comment raises the number of examples for this yet to be proven conjecture to at least 2, assuming you are not the same bored mathematician who inspired the video.
When I saw eigenvectors and eigenvalues flash up on screen it felt good remembering what those were from the little bit of linear algebra I did during college. I had a great professor. I kinda wish I had been going deeper into mathematics itself and not eventually shifted gears to computer science. He told us if we took his grad courses he would go into depth about the proof that got him his doctorate. Maybe I'll go back one day before he retires. I'm sure I could look it up but it hits differently when the person who discovered something explains it to you.
There was a question about eigenvalues on university challenge the other day, and I am ashamed to admit that I could only remember the term eigenvectors and that I had studied them at some point!
You can still learn some deeper mathematics. A good side project is writing an algebraic geometry program, just the simple stuff like declaring polynomial rings, irreducibility criterion, ideals, Grobner bases, stuff like that. It’s related to computer science and maths
Would love to see you do a video on the chess cheating controversy regarding Hans Niemann. The probabilities regarding this controversy are highlighted in Hikaru’s reaction video to the one Yosha posted. It might take a while to compile the data you would want, but it seems like something you would be interested in.
It would be a great example on how not to use probabilities
@@falquicao8331 It's because of the definition of top engine move. You can select a certain set of engines to stack the eval
no
"eigen see clearly now" omg i'm dying lmao
Oh wo that was awesome Matt!! Thank you so much
I love your song i listen it at bandcamp:)
I learned maths in German and for a moment I was like "Wait, they call it Eigenvektor, too?". Maths really is a universal language. :D
außer dass wir in deutsch die gottverdammten kommata verwenden statt einfach nen punkt
A lot of math terms are German, even in English. :)
The English word for Ansatz is amusingly also ansatz
@@timseguine2 Ich dachte, die englischsprachige Mathematikergesellschaft hat "Ansatz" bereits anglisiert, so wie es Matt mit "Eigen" (I can) bereits gemacht hat.
@@at7388 "onsotz"
“Square roots of 17, everywhere.” *stares into camera*
Math is hilarious if it’s well presented.
I stared at the puzzle at the start of the video for like 10 seconds before cracking it, and that euphoric high that came with the realization that I solved a puzzle in seconds, that an expert spent an entire day on cracking, is something I don't think is conceptually describable.
It's really stretching the definition to call Parker an "expert"
@@mrosskne He's a parker expert
@@mrosskne Let me have this one, my man
"We're not gonna do that using a network, we're gonna do that using a..."
SPREADSHEET!
"... matrix!"
OK I was close, right?
A spreadsheet is basically a matrix of cells
@@YOM2_UB They both have rows and columns, anyway
For those of you playing educational RUclipsr bingo, Matt mentioned the names of Crash Course and the English translation of Kurzgesagt (In a Nutshell), as well as 3Blue1Brown. Well done Mr. Parker, well done, even if the first two were simply expressions you used and not intentional, which, let’s be honest, they likely were.
YOU HAVE A BINGO CARD?? 🤯
It's gonna be a bright (bright) Smarter Every Day!
Matt: "A good reminder, just because it looks like something is the answer in mathematics, we don't know for certain until we do the maths"
I shall Burn this into mind, it is one of the many things I struggle with, I'm constantly trying to eye-ball things.
Amazing video.
You should become an engineer
Thanks Matt for the opportunity to look at you while you say things looking towards us, helps a lot!
For me, the most hilarious thing of all Matt's videos is to see how he primarily amuses himself with those Daddy maths jokes :D It totally cracks me every time :D
The way he smiles proudly smiles to himself right after his dad math jokes is perfect.
@@WreckedRectum Yeah, one of the better ones on rewatching was when he said "I hope you can, uh... matri-see? how eigenvalues and eigenvectors work". He kept a straight face for a split second before he just had to grin, and then he was so proud of the dad joke that he very slowly blinked. That slow blink was such a giveaway. I loved it!
I do remember seeing these kind of questions in maths books, and I learned to hate the very sight of them. Because they would appear, a bunch of stings of random numbers. And with no hint of any solution it just says "find the next number in the sequence". And in the answers it would be just a string of equally random seeming numbers. No hint of how one should arrive at that particular one. What the criteria for success was or anything. The result was that those who decrypted these hieroglyphic things felt smart and everyone else, me included, felt stupid. And encountering enough of similar tasks with no guidance, and no pedagogical help to be found. I just lumped them together as "one of those". I never got the tools to crack them, I was just left there, abandoned. Just writing a random number because a fail at least made it go away.
This, and the fact that the maths books in general I encountered were seemingly written backwards. Making me do tasks before teaching me how to do them. Making me wonder if I was having a stroke, if I was having early onset dementia, then I turn the page and it explained the mechanics to solve the past few pages tasks.
Sorry about the rant, I just had flashbacks to how exclusionary maths can be at times. I think it's part of why I liked computer classes, they at least knew how to teach. And in general, I knew what a success state was.
I like finding out about math stuff nowadays. 15-20 years after those schoolbooks. I just wonder how many mathematical minds were lost, because of horribly written books.
Put 1 2/3 ±0.5% in the specs and call it a day.
saw that sequence many years ago, it has been one of my favorites ever since
It's interesting to look at the binary look-and-say sequences with other starting points.
The sequence starting with 0 → 10 → 1110 → ... looks a bit different (all end with 0, of course). You'll still get the same block graph, just without 1 and 11 [which are irrelevant anyways], so we'll expect to get the same ratio of ones and zeroes in the limit.
But not every starting point gives such a sequence, for example when starting with 111, we just stay at 111, so the limit would be 0 (or ∞, depending on the point of view).
Are there other sequences which don't grow forever?
I think that is done in the video though - if you pause at 5:06 for example, number 6 goes to 100 (which is 4 in binary) 1s and 1 0 ...
@@cmcaulay07 Not only can you "encode the digits with representations higher than the base", he shows it being done at 5:04 with 1110->11110 and 11110->100110
@@cmcaulay07
> "You can't encode the digits with representations higher than the base"
> binary
> "you have to either pick "one one, two ones (11101), or two ones, one one (10111)"."
I lol'd.
I doubt there's any other sequences that don't grow forever.
Conway's solution for the original only works because of a theorem that every sequence eventually decays into a sequence of the atoms, and from that it's easy to see that the sequence 22 is the only one that could possibly work. (Everything else decays into something longer (after a step or two), and nothing else decays into specifically 22.)
There's no such supporting theorem here - after all, 111 literally isn't an atom in this structure. But I still don't see anything else as possibly being able to work, and I'd even expect adding 111 as an atom to make an equivalent theorem to Conway's to hold.
22
It's interesting how much this video resonates with Sabine Hossenfelder's "Lost in Math", which is very much worth reading on the subject of elegance as a goal in math and physics.
This game doesn't really work in spanish, since the word for the number 1 is different than the word for "1 of something" ("uno" and "un").
So if you say "un uno" (kind of like "a 1") it's different that saying "uno uno" ("one one").
1:13 "But now we've got two ones, so the next one is 2 1s." Except it isn't. It's "2 1", not "2 1s". So you've got a similar discrepancy in English.
@@omp199 yeah, but I guess the algorithm is still coherent beyond language: every time you start a new row of numbers state two inputs : 1st how many 2nd of which number, then keep going..
@@AuroCords Yes. That's right.
As soon as I saw the graph I knew I'd be "blessed" to see matrices pop up. I remember having to deal with a similar problem when I was prototyping a game and ran into a similar problem relating to random walks which turned to markov chains, which turned to eigenvalue shenanigans. Cool math, but not exactly transparent to the people playing the game or immediately intuitive.
Yep! This matrix is actually a representation of the graph, called adjacency matrix
Any chance that Eigan See Clearly Now is gonna be released anywhere? 👀 I absolutely love it!
I can't wait to have this as my ringtone
I love it when you give a number sequence and people think there's some sort of advanced algebraic or geometric relationship then its just... nope... Numbers.
Number theory relationships also matter.
I did a lot of engineering courses, saw eigenvalues and eigenvectors until I could claim I know them by heart. I thought that that segment wouldn't teach me anything new, but it did. I finally made the mental connection on why we actually use them for solving systems of differential equations!
Thank you, Matt!
:O
@@Cloiss_ :O
Can you do a video on Hans Niemann allegedly cheating in the chess Sinqued cup? I would love a statistical breakdown the same way you did with Dreams mc world record
Great video! Small correction: the result at 9:20 contains a sign error. The eigenvalue is indeed -1, but the eigenvector should therefore be [1 -1]
There's not a unique eigenvector corresponding to an eigenvalue. You can scale an eigenvector by a constant and it's still an eigenvector for the same eigenvalue. In fact, you can take any vector from the eigenvalue's eigenspace.
Very interesting video!
A small note: your audio is quite quiet, I had to max my speakers to reach a "normal" volume.
Maybe 2x as loud would be perfect?
Appreciate you!
Yes, it's very quiet, making the music and sound effects become extremely loud. Voice should be normalised to -6 dB.
So glad eigen finally see clearly now. Also, let it be known I did learn about eigenvalues in my Linear Algebra course!
That's so crazy. I was reading about quantum entanglement the other day and stumbled upon eigenvalues/ -vectors as they relate to the wave function collapse.
"Everything is, like, connected, man."
I want egian see clearly now full song. That be my bop of the year
I literally just finished studying eigen vectors and values like, a week ago (although it wasn’t called like that in my course)! What a coincidence! I wonder if Maple uses the same algorithm for finding these values as we used, one of the task was to write a program that finds them. There is a way to find two pairs of answers, to be specific, to derive one pair from another, which is really neat!
We really need a full version of "Eigen see clearly now"!
The Look and Say sequence will be added to my party tricks collection along with tying my shoes the mathematical way. Cheers
I'm unaware of the mathematical way of tying my shoes. Can you teach me?
The ultimate nerds and geeks party trick! I'd be at that party.
I'd like to know about the shoes thing too please
@@Starwort see the link above
@@annleach7908 there's no link, you got shadowbanned
FINALLY. After ALL THE YEARS, I understood Eigenvektoren and Eigenwerte.
I thought the motto of the channel was "Give it a go"
Well, I can say that I definitely wasn't expecting to have flashbacks to my linear algebra course today.
Thank you Simon, from Cracking the Cryptic, I got the sequence immediately.
@0:19 It's a read number. starting at 1. 1; one 1; two 1's; one 2, one 1; one 1, one 2, two 1's; three 1's, two 2's, one 1. Next: one 3, one 1, two 2's, two 2's. I'll continue watching in case there's an alternate sequence, but this one fits so far. The word one looks so weird right now. @1:00 didn't know there was a name for it. I've seen enough to make the comment.
I realized where you were going about 15 seconds before you said eigenvectors, and I was so proud, I'm not a mathematician
I got my monthly dopamine allowance by hearing you say the intro problem took all day when I solved it in ~15 seconds
Literally me yelling eigenvalues at my phone...also this thumbs up is for eigen see clearly now.
Hey, I did eigenvectors and eigenvalues in economics! We did it for one module, for one exam, and I've completely forgetten everything about them ever since.
I think it's amazing that someone was able to completely solve this problem, but we then have problems like the Collatz conjecture or the Lychrel number problem that can't be solved.
Neat description of Eigenvectors and Eigenvalues
Really great! The song is awesome! 😍
Look and say we're seeing saying13112221. When you take this sequence out to Infinity you still only ever see the integers 1 2 and 3 no other integers appear in the sequence. Also if you create a sequence that counts the amount of digits in each of the numbers of the look and say sequence and then create a ratio of two consecutive numbers in this new sequence as it approaches Infinity it equals a constant that is the unique positive real root of a polynomial of degree 71. The constant is called Conway's constant and it is approximately 1.303577...
"I spent all day but I cracked it"... it took me 20 seconds...
Matt Punner's Parks-- I mean Matt Parker's puns are always so out of nowhere but so good. Eigen See Clearly Now....
You made it to 1 million subscribers. Congratulations Matt :)
I felt like I was having a stroke during this explanation. I'm too old to have learned matrices in school, so this whole video felt like a definition of terms using only the terms defined within the definition. "The animal a dog is an animal that is a dog, and excludes animals that are not dogs. Even animals that are similar to dogs, but not dogs, are animals that are not dogs. Only dogs are dogs. Here is a proof."
To calculate the probability of an arbitrary elegant conjecture being true, we need to find the set of all elegant conjectures, and divide it into True, False, Unknown, and Not Well Defined.
A strict measure of Elegance will likely involve how many words or ideas are needed to describe the conjecture.
If we pick a maximum number of words at say 20, and look at all (words)^20 items, we must note the are more conjectures than can be examined in a reasonable amount of time.
Thus the conjecture 'elegant conjectures are trust-able' is in the Unknown category, for at least the age of one universe.
God I'm such a nerd for enjoying that song so much
It's delightful, isn't it? 😊
I faceplanted so hard when you revealed the answer to that sequence
As a German I shivered at the joke with "Eigen see clearly now."
9:21 shouldn't the resulting vector be [1,-1] with the eigenvalue being -1? At the moment that's flipping the sign repeatedly rather than showing the actual eigenvalue
Yeah, I agree with you. I slept poorly and thought my brain was just not working today, so thanks for the comment.
Yes, the eigenvector is [1, -1], but it's written as -1*[-1, 1]. Which is the same vector.
the best look and say sequence is 22,22,22,22,22,22,22...
I love that the look & say sequence feels like a pun that works in every(?) language
i dont know whats worse, the pun in the song or the pun in the card for Johny's channel at the end😭
Great video. And a nice refresher of Eigenvector and Eigenvalue. Thought about how to do this about a week ago
There was an 'Eigenmode' song at one of the Fermilab Physics Slams. I recall that it won over other entries.
This video immediately brought back my Data Structures and Algorithms PTSD
I felt kinda good solving it in 3 minutes of paused time when Matt said it took him a day.
It's the little things.
Got it in five seconds! Thanks for that!
What I find more interesting about this sequence (the base 10 one) is the question: "can a 4 or larger digit appear in the sequence?".
It is easy to get the number 3, for example ...12... -> ...1112... -> ...31...
In the first number I assume that it is proceeded by non-1 digit and followed by a non-2 digit. The second number must be proceeded by the same non-1 digit as the first number, but since we can not be sure how many non-2 digits were in the first number, we do not know whether the following digit is or is not a 2. This means that in the third number all we know for sure is that there are 3 1's in a row, but it shows that it is possible for a 3 to exist in a number in the sequence.
If you start the sequence with a digit larger than 3 it will be replicated in all subsequent numbers, but can you get one to spontaneously appear as a result of the sequence operation?
In order for a digit larger than 3 to spontaneously appear as a result of the sequence operation, the prior sequence must look something like
(1) BBBB... or (2) ...ABBBB... where I use ... to denote "the rest" of the sequence leftwards or rightwards.
Could (1) or (2) have been produced by a sequence operation?
For (1) we would read it as B lots of B, followed by B lots of B. But that can't happen. That is the same as 2B lots of B.
For (2) it may be read differently depending on parity.
- A lots of B, followed by B lots of B, followed by B lots of ?
- ? lots of A, followed by B lots of B, followed by B lots of B
but in either case, we have a situation of X lots of B, followed by Y lots of B. Which is the same as (X+Y) lots of B, which is impossible.
Thus a digit larger than 3 can't spontaneously appear.
As far as I can tell : The only way to have the string "1111" appear in a result (so that 41 would appear in the following result), is to say "one 1, one 1". You will never say this; on a previous entry, you would have said "three 1s" or "two 1s". Similarly, you will also never say "two 2s, two 2s" or "two 3s, two 3s". I don't know how to prove this.
That matrix *is* the graph. Lots of graph algorithms have very pleasant formulations in terms of operations on their adjacency matrices, at least when you can fit the adjacency matrix in memory...
10:36: a parade of square roots of seventeen promptly parades accross the screen.
damn that's a brilliant way to use an ad break
OK that joke was more than I expected, you won
I'm surprised. I figured it out in just a few minutes. Really helped having them displayed in a column like that.
Oh man I feel like I figured that out by chance, or maybe it's my mind associating 11 with one-ty one, but I got a huge Smile when I figured the pattern out!
Bruh you killed me with that "Eigen see clearly" bit. Lmao this is why they call you stand up maths.
Spoilers for the intro! I have paused the video before any explanation was given, so this is just my best guess.
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each term in the sequence is a string representation of the term before it. 312211 aka “three 1s, two 2s, one 1” describes the previous string “111221,” which describes “1211” - “one 1, one 2, two 1s.” Thus, the next term in the sequence, following 312211, is “13112221”
The Look and Say numbers. A lovely sequence, popularized by Late John Conway.
I recognised that puzzle as I worked it out and won money in a pub quiz because of it 😁
As an 80s kid, my brain kept going to "What's our vector, Victor?"
For a good moment I thought the ratio was 1.46 and I was shocked you were comparing them so equally
Brilliant video. I especially liked 5:12 where my physics brain was screaming IT‘S GONNA BE A MARKOV CHAIN. And it was. What an amazing link between such seemingly different problems!
For anyone interested: go look up "stochastic matrix"
I'm immediately glad I DIDN'T sit down and think about it for more than a minute.
Lack of pausing time? Like we don’t have a pause button 😂😂
Thanks for reminding me about eigenvalues and eigenvectors. Haven't had to think about those maths since college, which was extremely novel to see appear in one of your videos.