>>11015327>>11015218A functional is something that eats shit and produces a single number from a field (read R).
A functional that eats vectors is called a form. For instance, volume is a form that eats several vectors and gives you a number - the volume of the parallelepiped formed by these vectors. Another example - a work form that eats a force vector and gives out a number - the work produced by the force along some direction.
You can represent a form as a row - if you matrix-multiply it by a column-vector, you indeed get a number.
A linear operator is a thing that eats a a vector living in one vector space and produces a vector from (possibly another) vector space.
A matrix is a representation of a linear operator in a fixed basis (read coordinate axes).
What do you need to get this matrix to become a functional? It eats vectors but produces vectors. But we know that forms eat vectors and produce numbers. Therefore, if the matrix eats a single vector AND a single form, it produces a number.
More generally, an object that eats both vectors and forms is called a tensor. It can be pictured as matrices stacked on top of each other.
Be aware this explanation is not rigorous, ideally you would read about exterior algebra and de rham cohomology to really appreciate forms. The MTW book is a good intro on tensors as applied to relativity.
