Proof of Cramer's rule with Dirac Delta Function

We present a new proof of Cramer's rule by interpreting a system of linear equations as transformation of $n$-dimensional Cartesian-coordinate vectors. To find the solution, we carry out the inverse transformation by convolving the original coordinate vector with Dirac delta functions and changing integration variables from the original coordinates to new coordinates. Our formulation of finding a transformation rule for multi-variable functions shall be particularly useful in changing a partial set of generalized coordinates of a mechanical system.


I. INTRODUCTION
Cramer's rule [1] is a formula for solving a system of linear equations as long as the system has a unique solution. There are various proofs available [2][3][4][5] and six different proofs of Cramer's rule are listed in Ref. [5].
Dirac delta function δ(x) is not a proper function but a distribution defined only through integrals [6][7][8]: where f (x) is a smooth function. The Dirac delta function δ(x − a) projects out the value of a function f (x) at a certain point x = a after integration: . This elementary property of δ(x) can further be applied to change the variable: ∞ −∞ dx f (x)δ(x − y) = f (y). The Dirac delta function is particularly useful in physics. One can make an educated guess that the change of variables can be made for a multi-variable integral by introducing the integration of products of Dirac delta functions whose arguments contain the transformation constraints.
In this paper, we derive Cramer's rule by making use of the Dirac delta function to change the variables of multi-dimensional integrals. The change of variables corresponds to a coordinate transform ′ = Ê , where = (x 1 , · · · , x n ) T and ′ = (x ′ 1 , · · · , x ′ n ) T are two n-dimensional Cartesian-coordinate column vectors and Ê = (R ij ) is the corresponding invertible transformation matrix. By changing the integration variables from x ′ k 's to x k 's in a trivial identity = n k=1 , we carry out the inverse transformation = Ê −1 ′ . To our best knowledge, this is a new proof of Cramer's rule.
Our formulation of finding a transformation rule for multi-variable functions shall be particularly useful in changing a partial set of generalized coordinates of a mechanical system and can further be extended to reorganizing the phase space of a multi-particle interaction in particle phenomenology. * chodigi@gmail.com † jungil@korea.ac.kr ‡ chyu@korea.ac.kr This paper is organized as follows. After providing a description of the notations that are frequently used in the remainder of this paper in Sec. II, we proceed with the derivation of Cramer's rule by making use of Dirac delta functions in Sec. III. In Sec. IV, we discuss implications and possible applications of our results.

II. DEFINITIONS
We define n-dimensional column vectors and ′ whose ith elements are the Cartesian coordinates x i and x ′ i , respectively: The two coordinates are related as x ′ i = n k=1 R ik x k : where the transformation matrix Ê is an n × n invertible matrix with known elements. Cramer's rule states that the ith element of is determined as where Det represents the determinant, x i is the ith element of , and Ê (i) ( ′ ) is the n × n matrix identical to Ê except that the ith column is replaced with ′ . We define the j × j square matrix Ê [j×j] whose every element is identical to that of Ê for j = 1, · · · , n: and Ê (k) [j] with j rows as We define the partial contribution of ′ [j] originated from the higher-dimensional coordinates x j+1 through x n as We denote D j by the determinant of Ê [j×j] : We denote Ê (i) [j×j] (Î) by the j × j square matrix identical to Ê [j×j] defined in (5) except that the ith column is replaced with a column vector Î with j elements: where Ê (i) [n×n] (Î) = Ê (i) (Î).

III. PROOF OF CRAMER'S RULE
In this section, we illustrate the proof of Cramer's rule with the aid of the Dirac delta function.
We multiply the unities to for i = 1 through n. Then we find that The multiple integration over x ′ 1 , · · · , x ′ n can be replaced with the integration over the variables x 1 , · · · , x n as where J is the Jacobian Then the integral (11) reduces into (14) The integration over x 1 in Eq. (14) can be carried out as where D 1 = Det[Ê [1×1] ] = R 11 and x 1 in is replaced with the expression in terms of x ′ 1 and x k for k = 2, · · · , n as By employing mathematical induction, it is straightforward to find that the integration over x j can be carried out as where Ê (i) [j×j] (Î) is defined in Eq. (9) and ′ ⊥[j] is defined in Eq. (7) that depends only on x j+1 through x n . After the multiple integrations over x 1 in Eq. (15) through over x j in Eq. (17), x i for i = 1 through j are expressed in terms of x ′ 1 , · · · , x ′ j and x k 's for k = j + 1, · · · , n as We repeat the same procedure until we reach j = n at which ⊥[n] vanishes: As a result, for i = 1, · · · , n, where ). This completes the proof of Cramer's rule (4).

IV. DISCUSSION
We have derived Cramer's rule for a solution of the system of linear equations, ′ = Ê , where and ′ are interpreted as n-dimensional Cartesian-coordinate column vectors in Eq. (2) and Ê is the corresponding transformation matrix. The introduction of a trivial identity (11) involving Dirac delta functions has leaded to a way to reach the systematic transformation of the original coordinates x i 's into new coordinates x ′ i 's in a straightforward manner resulting in the completion of the proof of Cramer's rule (4). To our best knowledge, this proof is new and, furthermore, could be useful to the pedagogical training of recursive application of integration of Dirac delta functions.
In an intermediate step, we have derived Eq. (18), where x i (i = 1, · · · , j ) is expressed in terms of x ′ i and x k (k = j + 1, · · · , n) with the transformation matrix Ê.
In order to find the solution, we have to know information for the elements of the submatrix Ê [j×j] as well as those of another j × (n − j) matrix, R ik . Equation (18) implies that a similar form to Cramer's rule holds in the change of a partial set of variables. It turns out that Cramer's rule is actually a special case for j = n of the general formula (18), which we call Cramer's rule for a partial set of variables or coordinates. This rule can be immediately applied to change a partial set of generalized coordinates of a mechanical system and can further be extended to, for example, reorganizing the phase space of a multi-particle interaction in particle phenomenology.