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.
- exact finite differences derivative
- exact derivatives on partitions
- exponential function on a partition
- discrete quantum mechanics
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
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 antisymmetric, tridiagonal matrix
where —it can be pure real or pure imaginary—, , and . This function is well defined for , with value . This matrix is interesting because, as we will see below, it represents a derivation on a partition. A rescaled matrix is defined as
where , and
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 ,
Strikingly, we recognize the determinant B as the Fibonacci polynomial of index [10, 11], i.e., B =F . Fibonacci polynomials are defined as
Since we have that B =F , and the recursion relationship for Fibonacci polynomials, we also have that
Then, the eigenvalues of the derivative matrix are and , where is the -th root of the -th Fibonacci polynomial, which is a polynomial of degree [10, 11].
The system of simultaneous equations for the eigenvector corresponding to , can be put in a form similar to the recursion relationship for the Fibonacci polynomials, i.e.,
This set of recursion relationships can be written as the matrix equation
where and . Thus
i.e., the -th component of the -th eigenvector is
For the case of the eigenvalue , we can rewrite Eq. (17) by noticing that if we let ( ), then for . This can be proved by induction method as follows. For , it is immediately verified. First, suppose that the equality holds for . Next, we compute the right-hand side of the equality for . Substituting in the expression for , and using the properties of the Fibonacci polynomials, we obtain
Therefore, according to Eqs. (17) and (18), the eigenvector for the eigenvalue takes the form , where is a normalization constant. We can take advantage of the normalization constant and write
with eigenvalue (in original scaling, i.e., the eigenvalue of the matrix ), is an arbitrary constant, and . This means that the exponential function is an eigenvector of the derivative matrix which is a global representation of the derivative on the partition . Recall that the exponential function is an eigenfunction of the derivative of functions of continuous variable.
The remain of the eigenvectors have eigenvalues equal to the negative of the roots of the -th Fibonacci polynomial , , and have the form
The vector that we will be interested on is the one which is the exponential function (19) with eigenvalue .
3. The matrix represents a derivation
Let us consider a partition, , , of equally spaced points of the interval , , with the same separation between them.
The rows of the result of the multiplication of the derivative matrix and a vector are
where and . We recognize these expressions as the second order derivatives of the function at the mesh points, but instead of dividing by twice the separation between the mesh points, there is the function in the denominator. This function makes it possible that the exponential function be an eigenvector of the matrix .
The values and extend the original interval to so that we have well defined the second order derivatives at all the points of the initial partition, including the edges of the interval. When is the exponential function, we have and , i.e., they are the values of the exponential function evaluated at the points of the extension.
Thus, we define finite differences derivatives for any function defined on the partition as
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 , and , then
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 , results in
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 instance, for the first two points, the second derivative is
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 , including the edge points. For the first point, we have
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 , and they reduce to the corresponding expressions for continuous variables.
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, . Note that the inverse at the local level is a bit different from the expressions obtained by means of the inverse matrix (see below) of the derivative matrix . When dealing with matrices there are no boundary terms to worry about.
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 . However, here, we have to deal with two values at each boundary.
3.8. The chain rule
The chain rule also has a finite differences version. That version is
is a finite differences derivative of with respect to , and the second factor approaches the derivative of with respect to
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 —and the finite differences derivative. We begin with the derivative of ,
Hence, the finite differences derivative of the product is
This is the finite differences version of the commutator between the coordinate and the finite differences derivative D. This equality will become the identity operator in the small limit, as expected. An equivalent expression is
This is the finite differences version of the commutator between coordinate and derivative; the right hand side of this equality becomes in the small limit, i.e., it becomes the identity operator.
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 [ ] which will represent the coordinate partition
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 limit is just
5. An integration matrix
Since the determinant of the derivative matrix is not always zero, we expect that there exist an inverse of it. At a local level, the inverse of the finite differences derivation is the summation as was found in Eq. (46). In this section, we determine the inverse of the derivative matrix, and we find that it is a global finite difference integration operation.
Once we know the eigenvalues and eigenvectors of the derivative matrix , it turns out that we also know the eigenvectors and eigenvalues of the inverse matrix, when it exists. In fact, the equality , with , imply that
The inverse matrix is
Its determinant is
This matrix represents an integration on the partition, with an exact value when it is applied to the exponential function on the partition. When applied to an arbitrary vector , we obtain formulas for the finite differences integration, including the edge points
where . These are new formulas for discrete integration for the exponential function on a partition of equally separated points with the characteristic that it is exact for the exponential function .
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 . The relationship between discrete vectors on a partition and functions with a continuous argument makes use of continuous and discrete Fourier type of transformations, a wavelet . If we have a function of continuous argument , a conjugate vector on the partition is defined through the type of continuous Fourier transform as
and vice-versa, a continuous variable function is defined with the help of a discrete type of Fourier transform as
Assuming that the involved integrals converge absolutely, we can say that
The function is an approximation to the Kronecker delta function . The function shi is the hyperbolic sine integral shi . A plot of it is shown in Figure 1.
The ratio of sin functions, in this expression, is an approximation to a series of Dirac delta functions located at , . Thus, the operations and are finite differences inverse of each other.
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
Therefore, the discrete Fourier transform of the derivative of a vector is times the discrete Fourier transform of , plus boundary terms.
The Fourier transform of the derivative of a continuous function of variable is easily found if we consider the equality
The integration of this equality with appropriate weights gives
Hence, as is usual, the Fourier transform of the derivative of a function of continuous variable is equal to times the Fourier transform of the function, plus boundary terms.
We proceed with a brief discussion of the relationship between the derivative matrix
and an important concept in quantum mechanics; the concept of
In the finite-dimensional complex vectorial space (where each vector define a sequence
of complex numbers such that
). A transformation
is usually called
Let and be the eigenvalue of , where is a free parameter, the corresponding eigenvalue of is indeed the real value ; which is one of the properties of a Hermitian matrix, as is also the case of infinite-dimensional space (for the Hilbert space on a finite interval, these values are discrete, and for the Hilbert space on the real line, these values conform the continuous spectrum, instead of discrete eigenvalues). Other characteristic of is that the eigenvector corresponding to is the same exponential function which is the eigenfunction of (see Section 2).
Furthermore, let denote the adjoint of . Thus, if we restrict our attention to the off-diagonal entries , it is fulfilled that (noticing that, with then ). Even more, if we do not care about the two entries for , we will have a Hermitian matrix. Finally, as it was seen in Section 4, we can say that can be considered as a suitable approximation to the conjugate matrix to the coordinate matrix.
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.
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