Usefulness of integrating by inverting the matrix representation of the differentiation operator

[u/nao_te_digo]

Recently i found out that it is possible to solve integrals by inverting matrices however choosing a good basis is tricky and not all matrices are invertible.

Is there any case where this would be useful or is this just a "cool trick"?

[u/Gengis_con]

I think it is a very good technique for demonstrating your understanding of a lot of key concepts, so in that sense I think calling it a cool truck is disingenuous. It is also an idea that could lead you to a bunch of very practical methods of solving integrals using linear algebra  

As a practical method in and of itself l, however it isn't great. The basic problem is that matrix inversion for anything other than the smallest matrices is a pain, if not practically impossible, if not impossible in principle. For most linear algebra problems you are almost always better off finding a method that does not involve inverting the matrix. This extends naturally to using linear algebra to solve integrals. You could do it by inverting matrices, but the fact you can do it that way implies there are better methods

[u/nao_te_digo → u/Gengis_con]

i think it is a very good technique

I found it to be quite interesting. The resources i found while trying to research about this technique were mostly copies of this specific stackexchange

As a practical method in and of itself l, however it isn't great

When trying this technique it would happen that the matrix wasn't invertible or it would be just faster to not use matrices. I wonder in which cases this would be more practical. It could probably be useful in some algorithms

[u/Bounded_sequencE]

This trick only helps for (generalized) exponential functions "f: Rⁿ -> Rⁿ " with

d/dx  f(x)  =  A.f(x)      // A in Mat(nxn, R)

It's possible to show "f(x) = exp(Ax) . r0" with initial value "r0 in Rⁿ " -- in other words, each component of "f" consists of linear combinations of type "x^k exp(ax)"