A standard question of Gel'fond's, when listening to a talk, is: "What is the simplest nontrivial example?" This article shows how useful this principle can be.
Example 1: Counting and the Lagrange Identity
We will analyze the case for n=3. It will provide a beautiful outline for a general proof.
The left hand side of the equality (1) will be equal to
It would be helpful indeed to consider the following 3 x 3 matrix for the sum of all the elements of this matrix is equal to the LHS of (2).
In fact, the elements of the matrix are the terms on the RHS in (2).
Each term of the matrix above can thus be represented as
Now consider the first term on the RHS in (1). For n=3, it is
Again, as above, it would be helpful to consider the following matrix.
The elements of this matrix are again the terms on RHS in (3).
We can thus call each of them
The last term in (1) is
This time we use two matrices to organize the terms in (4).
And we thus define elements of the first and the second matrix as and respectively where
and where is defined to be if i=j and is equal to unity otherwise.
where we have used ( is the Kronecker delta)
Since each is zero, the sum
which automatically implies the Lagrange Identity in .