MATH
21 B
Mathematics Math21b Spring 2017
Linear Algebra and Differential Equations
Exhibit: Pascal Matrices
Course Head: Oliver Knill
Office: SciCtr 432

Pascal Matrices

In the Final exam, we looked at Pascal triangle matrices. So, here is the Pascal triangle, then the Pascal triangle, in which the integers are seen modulo 2 (we look at the remainder when dividing by 2, which means that we put 1 if the number is odd and 0 if the number is even). It was generated with the following line:
M=9; A=Table[ Binomial[n,k],{n,0,M-1},{k,0,M-1}]; Mod[A,2] 
1  0  0  0  0  0  0  0  0
1  1  0  0  0  0  0  0  0 
1  2  1  0  0  0  0  0  0 
1  3  3  1  0  0  0  0  0 
1  4  6  4  1  0  0  0  0 
1  5 10 10  5  1  0  0  0  
1  6 15 20 15  6  1  0  0 
1  7 21 35 35 21  7  1  0 
1  8 28 56 70 56 28  8  1 
1  0  0  0  0  0  0  0  0
1  1  0  0  0  0  0  0  0
1  0  1  0  0  0  0  0  0
1  1  1  1  0  0  0  0  0
1  0  0  0  1  0  0  0  0
1  1  0  0  1  1  0  0  0
1  0  1  0  1  0  1  0  0
1  1  1  1  1  1  1  1  0
1  0  0  0  0  0  0  0  1
Here is a larger version for n=29 = 512. The picture of the matrix converges to a fractal called the Sierpinski triangle.
In the exam, we have computed the first column of the inverse of the Pascal matrix. Here is a picture of the inverse for n=2048. The inverse takes values 0,1,-1 only.
Please send questions and comments to knill@math.harvard.edu
Math21b Harvard College Course ID:110989| Oliver Knill | Spring 2017 | Department of Mathematics | Faculty of Art and Sciences | Harvard University, [Canvas, for admin], Twitter