2

I have a linear operator with its matrix in certain coordinates to be

$$ \begin{pmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & \frac{1}{2} & 0 & \cdots & 0 \\ 0 & 0 & \frac{1}{3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \\ 0 & 0 & 0 & \cdots & \frac{1}{n} \end{pmatrix} $$

What is this linear operator? How could I construct it without referring to coordinates?

Yrogirg
  • 3,759

1 Answers1

5

Of course it could be any number of things, but one operator with this matrix is the one that assigns to every polynomial $p(x)$ of degree less than $n$ the polynomial $\frac1x\int_0^xp(t)\,\mathrm dt$.

joriki
  • 242,601
  • That's exactly how I came to this operator in connection with http://math.stackexchange.com/questions/142941/antiderivative-of-polynomials I just hoped that there is some other interpretation I could rely on. – Yrogirg May 09 '12 at 19:09