Polynomials, Matrices and Pascal Arrays | Algebraic Calculus One | Wild Egg

We introduce some basic orientation towards polynomials and matrices in the context of the Pascal-type arrays that figured in our analysis of the Faulhaber polynomials and Bernoulli numbers in the previous video. The key is to observe some beautiful factorizations that occur involving diagonal matrices as well as Toeplitz matrices which are constant on diagonal bands on and below the main diagonal.

As we delve further into the Algebraic Calculus, the algebra of polynomials and matrices will become increasingly important, and we will have to reconfigure those topics in terms of Vexels and Maxels, as well as polynumbers. So this video can be viewed as motivation for these developments.

Video Contents:
0:00 Intro
0:13 Numbers, Polynomials, Matrices
1:46 Traditional polynomials
3:20 Non-comutative arithmetic
5:57 Pascal arrays as matrices
11:22 Formulas
14:44 Inverses
19:02 Faulhaber's formula and Bernoulli numbers

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[JoelSjogren]

Lots of numbers. I see a familiar "function" lurking in the slides. Great video!

[jehovajah]

We start with the seemeia( indicators/ objects of independent interest). We sequence them, imposing a relational order on them, then we sum them

[douggwyn]

I=J^0 and it is interesting that the dimension of I is inherited from J.

[jehovajah]

Summing them imposes a series relation on the products . This then is our accounting, our calculus by synthesis.

[florankacaku]

Is there a general linear algebra theorem about when a matrix can be factored like this? Diagonal * upper triangular * diagonal

[dmitryezhov]

About variables. Calculus is mathematics of CHANGES! So the syntax of variables is very important. Standart syntax "f(x)" (where "f" and "x" is variables) - is very convenient and very-very simple. To see the full picture of the changes in polynomial functions, we need a complete syntax of the variables. It is necessary to see two states in a variable. Standart variable "x" - is change from ZERO to value of "x". It is not full syntax. The full syntax of variable - is change from BEGIN STATE to END STATE! Therefore the variable must be described by two-simbolic notation. The change from x1 state to x2 state, "x2-x1" - it is full syntax. "x2-x1" - is variable by two-simbolic notation. Yes, this is unusual and inconvenient, but maximal full info by change in variable. If the polynomial functions are described by two-symbolically variables, then it is easy to see how the binomial coefficients from Pascal array naturally arise in the derivatives, and multinomial coefficients in the derivatives of n-dimentional function. Yes, maybe it's "sensual" Calculus. If you do not feel and do not see two-states of variables in polinomial function, then it is difficult to understand this.

[postbodzapism]

on slide 9 why is there a missing summand of J^3?