What is the inverse of the above matrix ? I would strongly suggest that you think about the above matrix and what its inverse would look like before you read through.
On the face of it, it is indeed startling to even think of an inverse of an infinite dimensional matrix. But the only reason why this matrix seems weird is because I have presented it out of context.
You see, the popular name of the matrix is the Differentiation Matrix and is commonly denoted as .
The differentiation matrix is a beautiful matrix and we will discuss all about it in some other post, but in the this post lets talk about its inverse. The inverse of the differentiation matrix is ( as you might have guessed ) is the Integration Matrix ()
And it can be easily verified that , where is the Identity matrix.
Lesson learned: Infinite dimensional matrices can have inverses. 😀
In this series of posts about Legendre differential equation, I would like to de-construct the differential equation down to the very bones. The motivation for this series is to put all that I know about the LDE in one place and also maybe help someone as a result.
The Legendre differential equation is the following:
We will find solutions for this differential equation using the power series expansion i.e
We will plug in these expressions for the derivatives into the differential equation.
** Note: Begin
Let’s take .
As n -> . , -> .
As n -> , -> .
Again performing a change of variables from to n.
** Note: End
(iii) can now be written as follows.
x = 0 is a trivial solution and therefore we get the indicial equation:
We get the following recursion relation on the coefficients of the power series expansion.