The Derivative and the Integral as Infinite Matrices
The Derivative and the Integral as Infinite Matrices
A polynomial can be encoded as a vector using the coefficients of as the entries of . In this Demonstration column vectors are shown using round parentheses (like these) and row vectors using braces {like these}.
p
v
p
v
The vector space of of polynomials with coefficients over a field (like the real or complex numbers) has the infinite canonical basis .
{1,x,,,…,,…}
2
x
3
x
n
x
The derivative and the integral on are linear transformations. Their infinite matrix representations have nonzero entries above or below the main diagonal. Imagine them as filling the fourth quadrant, with dimensions ∞×∞.
D
J
D
0 | 1 | 0 | 0 | ⋯ |
0 | 0 | 2 | 0 | ⋯ |
0 | 0 | 0 | 3 | ⋯ |
0 | 0 | 0 | 0 | ⋯ |
⋮ | ⋮ | ⋮ | ⋮ | ⋱ |
J
0 | 0 | 0 | 0 | ⋯ |
1 | 0 | 0 | 0 | ⋯ |
0 | 1 2 | 0 | 0 | ⋯ |
0 | 0 | 1 3 | 0 | ⋯ |
⋮ | ⋮ | ⋮ | ⋮ | ⋱ |
The matrix product is the infinite identity matrix, but has a zero in the top-left spot. In a finite-dimensional vector space, if and are square and , then and is the unique inverse of . Linear transformations on infinite-dimensional vector spaces can be both familiar and strange!
D.J
J.D
A
B
A.B=I
B.A=I
B
A