# What is this linear operator/matrix?

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?

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linear operator it is kernel yes? –  dato datuashvili May 9 '12 at 19:02
i think it is null space ,or all set of vectors x,for which $A*x=0$ –  dato datuashvili May 9 '12 at 19:04
@dato: this matrix is full rank –  Alex R. May 9 '12 at 19:07
yes i see determinant is not zero,but how it is related to linear operator? –  dato datuashvili May 9 '12 at 19:09
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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 $\frac1x\int_0^xp(t)\,\mathrm dt$.

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i did not know it @joriki,what is in this case polynomial?matrix or? –  dato datuashvili May 9 '12 at 19:07
@dato: It's nothing special in this case, just a plain vanilla polynomial in one variable, the kind you're likely to meet in the street on your way to the bus stop. –  joriki May 9 '12 at 19:08
That's exactly how I came to this operator in connection with math.stackexchange.com/questions/142941/… I just hoped that there is some other interpretation I could rely on. –  Yrogirg May 9 '12 at 19:09