## Abstract

We introduce and study a matrix which has the exponential function as one of its eigenvectors. We realize that this matrix represents a set of finite differences derivation of vectors on a partition. This matrix leads to new expressions for finite differences derivatives which are exact for the exponential function. We find some properties of this matrix, the induced derivatives and of its inverse. We provide an expression for the derivative of a product, of a ratio, of the inverse of vectors, and we also find the equivalent of the summation by parts theorem of continuous functions. This matrix could be of interest to discrete quantum mechanics theory.

### Keywords

- exact finite differences derivative
- exact derivatives on partitions
- exponential function on a partition
- discrete quantum mechanics

## 1. Introduction

We are interested on matrices which are a local, as well as a global, exact discrete representation of operations on functions of continuous variable, so that there is congruency between the continuous and the discrete operations and properties of functions. Usual finite difference methods [1, 2, 3, 4] become exact only in the limit of zero separation between the points of the mesh. Here, we are interested in having exact representations of operations and functions for *finite* separation between mesh points.

The difference between our method and the usual finite differences method is the quantity that appears in the denominator of the definition of derivative. The appropriate choice of that denominator makes possible that the finite differences expressions for the derivative gives the exact results for the exponential function. We concentrate on the derivative operation, and we define a matrix which represents the exact finite difference derivation on a local and a global scale. The inverse of this matrix is just the integration operation. These are interesting subjects by itself, but they are also of interest in the quantum physics realm [5, 6, 7].

In this chapter, we will consider only the case of the derivative and the integration of the exponential function.

## 2. A matrix with the exponential function as an eigenvector

Here, we consider the

where

where

We are mainly interested in finding the eigenvalues and the corresponding eigenvectors of these matrices.

We start our study with a result about the determinant of

where

and

Strikingly, we recognize the determinant B

Since we have that B

and then

Then, the eigenvalues of the derivative matrix

The system of simultaneous equations for the eigenvector

This set of recursion relationships can be written as the matrix equation

where

but

and then,

i.e., the

For the case of the eigenvalue

Therefore, according to Eqs. (17) and (18), the eigenvector for the eigenvalue

with eigenvalue

The remain of the eigenvectors have eigenvalues equal to the negative of the roots of the

The vector that we will be interested on is the one which is the exponential function (19) with eigenvalue

## 3. The matrix D N represents a derivation

Let us consider a partition,

The rows of the result of the multiplication of the derivative matrix

where

The values

Thus, we define finite differences derivatives for any function

to be used on the first, central, and last points of the partition.

The determinant of the derivative matrix is not always zero, and in fact, it is [see Eqs. (4) and (9)]

But, since

Hence, only the matrices with an odd dimension have an inverse.

Next, we will derive some properties of these finite differences derivatives.

### 3.1. The derivative of a product of vectors

There are two equivalent expressions for the finite differences derivative of a product of vectors defined on the partition. A set of such expressions is

A second set of equalities is

### 3.2. Summation by parts

The sum of Eqs. (28) or (31), with weights

or

This is the discrete version of the integration by parts theorem for continuous variable functions, a very useful result.

### 3.3. Second derivatives

Expressions for higher order derivatives are obtained through the powers of

For inner points we get

and for the last two points of the mesh, we find

These derivatives also have the exponential function as one of their eigenvectors, and we can generate expressions for higher derivatives with higher powers of the derivative matrix.

### 3.4. The derivative of the inverse of functions

It is possible to give an expression for the derivative of

For central and last points, we find that

The derivatives for the first and last points coincide with the derivative for central points when

### 3.5. The derivative of the ratio of functions

Now, we take advantage of the derivative for the inverse of a function and the derivative of a product of functions and obtain what the derivative of a ratio of functions is

expressions which are very similar to the continuous variable results. Again, these expressions coincide in the limit

### 3.6. The local inverse operation of the derivative

The inverse operation to the finite differences derivative, at a given point, is the summation with weights

This equality is the equivalent to the usual result for continuous functions,

### 3.7. An eigenfunction of the summation operation

Because the exponential function is an eigenfunction of the finite differences derivative and according to Eq. (46), we can say that

in agreement with the corresponding continuous variable equality

### 3.8. The chain rule

The chain rule also has a finite differences version. That version is

where

is a finite differences derivative of

Thus, we will recover the usual chain rule for continuous variable functions in the limit

## 4. The commutator between coordinate and derivative

Let us determine the commutator, from a local point of view first, between the coordinate—the points of the partition

Hence, the finite differences derivative of the product

i.e.,

This is the finite differences version of the commutator between the coordinate

This is the finite differences version of the commutator between coordinate and derivative; the right hand side of this equality becomes

### 4.1. The commutator between the derivative and coordinate matrices

The commutator between the partition and the finite differences derivative can also be calculated from a global point of view using the corresponding matrices. Let the diagonal matrix [

Then, the commutator between the derivative matrix and the coordinate matrix is

This is a kind of nearest neighbors’ average operator, inside the interval. The small

where **I** is the identity matrix, with the first and last elements replace with 1/2. Thus, coordinate and derivative matrices are finite differences conjugate of each other.

## 5. An integration matrix

Since the determinant of the derivative matrix

Once we know the eigenvalues and eigenvectors of the derivative matrix

The inverse matrix

Its determinant is

This matrix represents an integration on the partition, with an exact value when it is applied to the exponential function

where

## 6. Transformation between coordinate and derivative representations

Since one of the eigenvalues of the derivative matrix is a continuous variable, we can talk of conjugate functions with a continuous argument

and vice-versa, a continuous variable function is defined with the help of a discrete type of Fourier transform

Assuming that the involved integrals converge absolutely, we can say that

where

The function

Additionally,

where

The ratio of sin functions, in this expression, is an approximation to a series of Dirac delta functions located at

### 6.1. The discrete Fourier transform of the finite differences derivative of a vector

Next, based on Eq. (28), we find that

If we sum this equality, we get

i.e.,

Therefore, the discrete Fourier transform of the derivative of a vector

The Fourier transform of the derivative of a continuous function of variable

The integration of this equality with appropriate weights gives

i.e.,

Hence, as is usual, the Fourier transform of the derivative of a function

## 7. Conclusion

We proceed with a brief discussion of the relationship between the derivative matrix *self-adjoint operators* [8, 9]. In particular, we focus on the momentum operator, whose continuous coordinate representation (operation) is given by

In the finite-dimensional complex vectorial space (where each vector define a sequence *Hermitian*, when its entries

Let

Furthermore, let

In conclusion, we have introduced a matrix with the properties that a Hermitian matrix should comply with, except for two of its entries. Besides, our partition provides congruency between discrete, continuous, and matrix treatments of the exponential function and of its properties.