Advice for research mathematicians
We are advocating a larger view of number theory which goes from arithmetic with numbers to polynumbers to maxels. In this lecture we have a look at the Chebyshev polynumbers of the first kind from this larger linear algebraic point of view. Some surprises are in store!
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0:06
[Music] Hello everyone i'm Norman Wildberger thanks for joining me on this adventure we are
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learning about extending number theory to the maxel situation so i'm proposing this is a huge
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and beautiful new arena for amateur mathematicians to get involved with connecting with concrete
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beautiful and i think important new mathematics and i want to illustrate the general idea by
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looking at very specific and important example the Chebyshev polynomials of the first kind
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i'm going to show you how to move from looking at these things from a one-dimensional point of view
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to a two-dimensional point of view where we bring in matrix theory linear algebra and some surprises
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are in store for us so let me remind you that the championship polynomials of the first kind
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defined recursively say t 0 of x equals 1 t 1 of x equals x and then t n of x equals 2 times
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x times t n minus 1 of x minus t n minus 2 x and here's some graphs of them they're particularly
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sort of interesting in this box from x equals minus 1 to plus 1 and y equals minus 1 to plus 1.
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now here is actually an explicit formula for the Chebyshev polynomials you can see that you can
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expand it out in a polynomial in x powers of x uh depending on n so n is the index which is a
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natural number and k is going from zero up to n over two or the floor of n over two that's the
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greatest integer less than n over 2 and k z over 2 is a fraction
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and the coefficients involved here are well these binomial coefficients n minus k
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choose k and we're dividing by n minus k so it's sort of a sub binomial coefficient you might say
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and there's also a sign okay and explicitly well one way of getting at it is through the generating
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function and the generating function is given here it's 1 minus tx over 1 minus 2tx plus t squared
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involving a dummy variable t and then if you sort of do the long division or get your computer to
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do the long division you get this expansion and powers of t and the various coefficients
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are polynomials x which are exactly the Chebyshev polynomials or polynomials and if we list them
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then we uh we get this list right here okay so 1x 2x squared minus 1 4x cubed minus 3x etc these
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are these various coefficients of the powers of t that you get by expanding the generating function
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so today we want to move to something new okay and it's this step here from going from this family of
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essentially one-dimensional objects namely these polynomials or polynomials to a two-dimensional
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family so this is what i'm sort of advocating is that is an interesting thing to think about
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interesting um sort of door to go through okay so let's have a look how do we do this well we think
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of each one of these polynomials as in this case say a horizontal string of coefficients
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starting with the lowest degree ones first so this 2x squared minus 1 is represented here in this row
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with a minus one and then a two and there's a gap we're kind of going every second one
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that's because the powers here are x to the zero and then x squared we're missing an x to
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the one in this one here and then the next one is um minus 3x plus 4x cubed there it is there
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there's that row minus 3x plus 4x cubed and so on so we're just listing them row by row and
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now we get this two-dimensional triangular let me spell get a triangular on maxim so a maxel is uh
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a more data theoretic sort of approach to matrices it's not really necessary that you're completely
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familiar with that just think in terms of a matrix that kind of goes on so we can extend
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this by going to the right and down okay and so rather arbitrarily just going up to in this case
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degree 10 or degree 9 or whatever okay but you should think of this kind of picture as just a
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snapshot of the top left corner of this this this object which in principle keeps going down and we
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don't have to make any grandiose claims about you know what happens at infinity or something it just
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keeps on going and ultimately realistically it'll overwhelm our ability to to represent
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it that's actually what really happens okay so what we're interested in doing now is asking
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the question is there any benefit to be gained by going from sort of this one-dimensional point of
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view having a lot of one-dimensional objects to having this single two-dimensional object
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well the answer to this sort of depends on whether we can apply matrix theory or linear algebra to
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this to this resulting maxillary matrix and more specifically that involves asking the
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question whether matrix multiplication or maybe even matrix inversion that's that's the crucial
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thing actually has something to say about this but the first step would be just to look at this
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thing as a as a two-dimensional array and see if we can you sort of notice anything about it
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that's maybe not so apparent from this kind of description where we just see the polynomials
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one at a time okay so have a look at this can you see anything about it that's interesting
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well one obvious thing to do which is sort of a first thing that you might think to do
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is just to factor the elements here and see what the kind of what kind of numbers are going on
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here this is a kind of a number theory so factoring is is great so this will factor
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all of the entries sorry simultaneously into their prime decompositions and if you sort of
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stare at this okay you see there's lots of twos there's some threes there's some fives and i guess
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about seven is about the greatest that you get so this is strongly suggesting that the formula for
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for these entries has something to do with factorials because the factorials are growing the
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fact that we're not getting any any higher prime factors here is a strong indication so even if you
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didn't know that there was a formula in terms of essentially a binomial coefficient you could uh be
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steered strongly to that you know by just staring at this factorization so that's a a very first
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important advantage in in looking at things this way is that you can sort of quickly
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see uh whether there's likely to be a nice factorial type formula for the entries of the
6:45
of the array in fact we do know that there's that formula the formula i already showed you before
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well here's uh something else that you might notice you might notice that okay like this
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column here that all the things are even here they're all odd here they're all even
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and then here they're all multiples of four these ones are all multiples of eight and this
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column here has all multiples of 16. you can check these are multiples of 32 and this is actually a
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multiple of 64. so suggesting that these columns are sort of multiples of powers of two also the
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powers of two are kind of obviously sort of strung along down the diagonal so that's you know another
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indication that might be quite something natural so one way of getting at that is to introduce
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um this matrix here okay
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let me move it up so you can see so let's introduce this matrix maybe i'll put it over here
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so let's let's give this thing first of all a name so the Chebyshev maxel let's call it on
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maxel is c and now let's just introduce this diagonal matrix with powers of two
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okay two to the zero to the one there's also a one up here that's maybe more questionable but okay
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we'll stick a one in there to make it invertible and then what we're going to do is we're going
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to sort of factor out these powers of two that are in the columns here and the way we do that
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is we compute c times d inverse so if you compute d inverses as a diagonal matrix so the the inverse
8:27
will just consist of these same diagonal entries but all of them inverted along the diagonal
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okay so when you multiply c by d inverse what's going to happen is you're going to be dividing
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each one of these columns by the corresponding power of two okay so you're going to simplify
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you're taking out those those powers of two and you can see that the result is in fact
8:51
simpler okay so this is now a simpler object and we've got it the matrix multiplication admittedly
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not a very complicated one because we've just been multiplying by a diagonal matrix but anyway
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we have in some sense sort of simplified our understanding of these chebychev polynomials
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by doing this right this is a a simpler kind of data structure than the original c up here was
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okay so um let's have a look at this thing here um can we say anything about this thing now
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well you see this one minus three minus four minus five minus six minus seven minus eight
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okay um you might think maybe that should be a two
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okay um but um anything else that we can obviously see about this not entirely clear
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all right so what's the next thing to do well
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let's call this the scaled Chebyshev maxim okay c times d inverse let's take its inverse
9:58
all right so the inverse of c d inverse this thing that we've just computed is d times c inverse
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okay now i'm strongly using the fact that we're in a matrix environment here and this is what i
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want to do because i want to check whether there's some linear algebra aspect to this
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situation and computing an inverse of a matrix is like a sort of an obvious thing that you should do
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if you have a matrix in front of you and you're kind of interested in it calculates determinant
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first and then if that's non-zero calculate the corresponding inverse that's telling you
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something important about the matrix so what's the inverse of this matrix here here it is right here
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notice that the minus ones have disappeared
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okay and what else has happened well it looks a little bit different from
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this one it still has this sort of alternating checkerboard aspect
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and if you just stared at this for a little while you might notice that the rows appear to be like
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pascal rows right one four three okay one five ten that's sort of like one five ten ten five one
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one six fifteen twenty fifteen 6 1. that's the sixth row of the pascal matrix except for
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there should be a 20 here okay um that's the start of the seventh one the eighth one 1 8 28 56 35 the
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next one really ought to be 70. so this is kind of cool it's it's making us think that this is almost
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like uh kind of a pascal matrix sort of staggered except that to get that we would have to multiply
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this first column by 2. what about this top entry here well that's just sort of there by itself
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maybe we could leave it where it is so if we double the entries of the first column except
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for the top entry i've just done that right so i've doubled all the entries of this first column
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so this is n then this new thing is let's say m and now i can observe that you know this is
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just a pascal matrix exhibited in a different fashion
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how can i show you that here it is here perhaps okay compare this m here with this matrix here
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so have a look at this matrix here what is this matrix here well it's a kind of a two-sided matrix
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doesn't just go off to the right it also goes off to the left okay so you should think about
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this as sort of being the center here okay and there's a 1 and then there's a 1 1 and 1 2 1 1 3
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3 1 so you see this is just a pascal triangle but sort of put symmetrically now so that the central
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binomial coefficients are right down the middle that's actually an interesting sort
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of symmetry now that the thing is completely symmetrical right along this sort of middle uh
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vertical axis okay so how does this thing compare with this
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well um it's just a truncation right it's what you get if you just look at that part of it okay
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let me just uh copy that and put it over here so i just copied that half of it or a little bit more
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than half because i've also included the central line and this is the same thing as that thing
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okay this is um i think quite cool so it's telling us that if we let's go back a little bit okay so
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we we started with the Chebyshev maxel where we just making a a matrix or a maxel out of the
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out of the Chebyshev polynomials now we have a two-dimensional thing then what we do is
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we sort of scale it to take out the obvious powers of two that are sort of complicating it okay
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so we take out those obvious powers of two and we get this matrix c d inverse which is just a scaled
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version of what we started with and this thing here it's inverse so now here's where we're
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using the crucial property the inverse of a matrix the inverse of this thing
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is is uh is this the n okay and then if we perform this somewhat curious modification to n
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by by um uh doubling the entries of this first column except for the top entry okay then this
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new matrix m that we get is immediately just a uh truncation or a part of a staggered pascal matrix
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so from this point of view you know if you started with the pascal matrix admittedly in this sort of
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unusual sort of symmetrical form then you could think about the Chebyshev Maxel as just something
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that you obtain from it by by taking the the right hand side and doing a little bit of a modification
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to that first column and then taking an inverse and then doing another scaling okay admittedly
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but the the essential sort of linear algebra component is is really contained already in
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this pascal matrix so that means that there's an intimate connection um with you know between
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the Chebyshev polynomials and and the binomial coefficients which of course you can see from
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the just the formula for the Chebyshev polynomials but this is a different aspect of it where we're
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we're utilizing the inverse structure of the maxels we're using the linear algebra
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of this two-dimensional situation to get at this inverse structure and arguably the infrastructure
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is simpler right easier to remember it's just this pascal matrix so we might say well what
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what happens if we if we go back so um so if this thing here is m okay and it's obtained
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from the staggered pascal matrix and if we um sort of you know um write this as d times c
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one inverse so so we sort of modify the original Chebychev polynomial of maxel from c
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to c1 then we would get that c1 would have to be m inverse d which turns out to be this thing so
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you know from some point of view sort of from this linear algebra or pascal point of view this maybe
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is a more natural kind of uh you know Chebyshev maxel you know from from that point of view
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maybe the thing that we started with is not exactly the best
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the best thing right this might be actually more interesting at least from some point
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of view at least from this point of view so let's investigate this one a little bit more
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so we can compare it with the original Chebyshev matrix so let's uh let me just do it so um
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here was um the original one okay and the c and here is the comparative if we're looking at sort
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of the poly numbers so the way to get the poly numbers very quickly from the the maxel is just
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to multiply by this uh column vector of powers of x that's basically just you know churning out
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the various polynomials corresponding to each row here okay so if we do that multiply by this
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vector of x's then we get the original list of Chebyshev poly numbers and if we factor them we
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can do that all at once okay you've seen me do this before so we factor them we get the
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factorization of the Chebyshev polynomials which we've noted before is is interesting but it's not
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quite as interesting or not at all as interesting as the as the expansion or the factorization of
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the spread polynomials which we will have to get to in a subsequent lecture but let us now compare
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this with this modified thing this c1 okay so this was this was the original c let me put that there
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c okay that's the original one and now this modified one so there's c one okay c sub one
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whoop c sub one and i've sort of multiplied it by x let me just do something like that just
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so c1 times this vector x so there's there's what you get and here then the sort of modified
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Chebyshev poly numbers and how do they compare with these ones well they're almost the same
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right because the only thing that we've really done differently is we've um we've modified the
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first uh column here so we've taken the first entries which are just the constant terms which
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are appearing in every second poly number right the ones that do have constant terms
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instead of having minus 1 plus 1 minus 1 plus 1 we're changing them
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into minus 2 plus 2 minus 2 plus 2. that's all the difference we're not noticeably changing
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the other ones so the these ones that don't have a constant term they're not being changed
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okay that's curious and now let me factor this thing here
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so let's compare this with compare with to get them side by side so
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here are the factorizations of the Chebyshev polynomials themselves
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here are the factorizations of the Chebyshev polynomials so on the right the original Chebyshev
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poly numbers on the left these modified Chebyshev quality numbers where we've just
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doubled the constant term um where where we were able to except for the top entry
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what's the difference well of course every second one is is is unchanged because they're
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not being changed but the ones that um that we are changing it appears at least
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from this relatively limited um examination that we're getting increased factorization
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we're getting increased factorization for example uh like let's say so here this um this fourth
20:29
Chebyshev polynomial okay 1 minus 8 x squared plus eight x to the fourth that doesn't factor
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but if you replace the one with a two like up here then you do get a factorization it's two times two
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x squared minus one all squared and similarly this one here we did have a factorization for
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it but now we get a richer factorization by just by adding or by doubling the the constant term
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and similarly here this one here didn't factor at all but over here we get a factorization
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does this continue i don't know i don't know if it continues i mean i'm willing to bet that it does
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but i don't know i don't have a proof
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what's the significance of these sort of modified Chebyshev polynomials well i don't
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know so the only thing i've sort of done is sort of graphed some of them like some of the these
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first ones you know and i get graphs that look like this you know there's quite a few of them
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here but you see that they're still bouncing around in a pleasant way but not quite as
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nicely constrained in in the in the minus one to one minus one to one box as the chebyshev
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polynomials were so what can we conclude so we can conclude that going to this two-dimensional
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maxwell point of view seems to be a rich thing for one thing it allows us to
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see that the chebychev maxwell its its inverse is intimately connected with the uh the pascal
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triangle arranged in a kind of a symmetrical matrix form sort of a staggered pascal matrix
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so that is definitely a linear algebra aspect of what's going on and moreover it suggests sort of
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these modified Chebyshev polynomials which very well may be uh interesting maybe more interesting
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from a number theoretic or factorizable sort of point of view okay so there's i think a
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great potential for study here to try to understand you know what's going on with
22:41
these things have these things been studied before i don't know can you say anything about
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whether these nice factorizations continue or not these are all interesting questions. [Music] you