Matrices with a Diagonal Commutator

Abstract

It is well known that there are no two matrices with a diagonal commutator. However, the commutator can behave as if it is diagonal when acting on a particular vector. We discuss pairs of matrices that give rise to a diagonal commutator when applied to a given arbitrary vector. Some properties of these matrices are discussed. These matrices have additional, continuous eigenvalues and eigenvectors than the dimension of the matrix, and their inverse also has this property. Some of these matrices are discrete approximations of the derivative and integration of a function and are exact for the exponential function. We also determine the adjoint of the obtained discrete derivative.

1. Introduction

Let us consider the matrices that shift the entries of a vector. The usual matrix that cyclically shifts the entries of a vector to the left is

$${\mathbf{R}}_1=\left(\begin{array}{cccccc}0& 1& 0& \dots & 0& 0\\ 0& 0& 1& \dots & 0& 0\\ 0& 0& 0& \dots & 0& 0\\ ⋮& & & & & \\ 0& 0& 0& \dots & 0& 1\\ 1& 0& 0& \dots & 0& 0\end{array}\right).$$E1

Given an arbitrary vector $\mathbf{h}={\left(h_1,h_2,\dots ,h_N\right)}^T\in \mathbb{C}$, $h_j\ne 0$, the action of ${\mathbf{R}}_1$ on this vector results in

$${\mathbf{R}}_1\mathbf{h}={\left(h_2,h_3,\dots ,h_N,h_1\right)}^T.$$E2

There is also a matrix that shifts the entries of a vector to the right, the matrix with the lower diagonal entries different from zero.

But, we can also use a diagonal matrix to rotate to the left the entries of the vector h, in particular. The action of the diagonal matrix

$${\mathbf{R}}_2=\left(\begin{array}{cccccc}\frac{h_2}{h_1}& 0& 0& 0& \dots & 0\\ 0& \frac{h_3}{h_2}& 0& 0& \dots & 0\\ 0& 0& \frac{h_4}{h_3}& 0& \dots & 0\\ 0& 0& 0& \frac{h_5}{h_4}& \dots & 0\\ ⋮& & & & & \\ 0& 0& 0& 0& \dots & \frac{h_1}{h_N}\end{array}\right)$$E3

on h is ${\mathbf{R}}_2\mathbf{h}={\left(h_2,h_3,\dots ,h_N,h_1\right)}^T$. It is also possible to perform a cyclic rotation to the right. The action of this matrix on another vector is only to rescale its entries.

Combining the above ideas gives rise to a third type of shifting matrix. There is a non-diagonal matrix that acts like an identity matrix for the particular vector h:

$${\mathbf{R}}_3=\left(\begin{array}{ccccccc}0& \frac{h_1}{h_2}& 0& 0& \dots & 0& 0\\ 0& 0& \frac{h_2}{h_3}& 0& \dots & 0& 0\\ 0& 0& 0& \frac{h_3}{h_4}& \dots & 0& 0\\ ⋮& & & & & & \\ 0& 0& 0& 0& \dots & 0& \frac{h_{N-1}}{h_N}\\ \frac{h_N}{h_1}& 0& 0& 0& \dots & 0& 0\end{array}\right).$$E4

We have that ${\mathbf{R}}_3\mathbf{h}=\mathbf{h}$. The eigenvalues of this matrix are the same as those of ${\mathbf{R}}_1$, the N roots of unity: ${\lambda }_n=e^{i2\mathit{\pi n}/N}$, $n=\mathrm{0,1,2},\dots ,N-1$, and h is the eigenvector that corresponds to the eigenvalue ${\lambda }_0=1$.

Other special matrices are the matrices that admit a continuous eigenvalue, besides the usual constant eigenvalues. For instance, the matrix

$${\mathbf{Q}}_1=\left(\begin{array}{ccc}{q}_{11}& q_{12}& 0\\ 0& q_{22}& q_{23}\\ 0& q_{32}& q_{33}\end{array}\right)$$E5

has eigenvalues $q_{11},\frac{1}{2}\left(q_{22}+q_{33}-z\right),\frac{1}{2}\left(q_{22}+q_{33}+z\right)$, and the corresponding eigenvectors are

$${\left(\mathrm{1,0,0}\right)}^T,$$E6

$${\left(\frac{q_{12}\left(-q_{22}+q_{33}+z\right)}{q_{32}\left(2q_{11}-q_{22}-q_{33}+z\right)},-\frac{-q_{22}+q_{33}+z}{2q_{32}},1\right)}^T,$$E7

$${\left(\frac{q_{12}\left(q_{22}-q_{33}+z\right)}{q_{32}\left(-2q_{11}+q_{22}+q_{33}+z\right)},\frac{q_{22}-q_{33}+z}{2q_{32}},1\right)}^T,$$E8

where, $z=\sqrt{4q_{23}{q}_{32}+{\left(q_{22}-q_{33}\right)}^2}$. These are the only eigenvalues and eigenvectors. The eigenvalues are fixed quantities and depend on the fixed entries of the matrix. However, the matrix

$$\left(\begin{array}{ccc}\frac{v_1\tau }{v_1-v_2}& -\frac{v_1\tau }{v_1-v_2}& 0\\ 0& \frac{v_2\tau }{v_2-v_3}& -\frac{v_2\tau }{v_2-v_3}\\ 0& -\frac{v_3\tau }{v_3-v_2}& \frac{v_3\tau }{v_3-v_2}\end{array}\right),$$E9

has eigenvalues 0, $\tau$, $\frac{v_1\tau }{v_1-v_2}$ with corresponding eigenvectors

$${\left(\mathrm{1,1,1}\right)}^T,{\left(v_1,v_2,v_3\right)}^T,{\left(\mathrm{1,0,0}\right)}^T,$$E10

where the eigenvalue $\tau$ is independent of $v_j$, and viceversa, and it is a continuous variable.

There is the usual procedure to obtain a set of eigenvalues and eigenvectors [1]. But if the entries of a matrix contain variables, we can solve the eigenvalue set of simultaneous equations now for the entries of the matrix, obtaining additional continuous eigenvalues and eigenvectors.

Now, in general, there are no two matrices that have a diagonal commutator [2], that is, proportional to the identity matrix. We use the above facts about matrices to define two matrices with a diagonal commutator when applied to an arbitrary vector h. One of the obtained matrices corresponds to a finite differences derivation of a function, exact for the exponential function.

2. The matrices

There are many matrices with a diagonal commutator along a given direction, but we consider a simple set for simplicity.

We consider the pair of N×N, $N\in \mathrm{\mathbb{N}}$, matrices A, with non-zero elements around and on the diagonal, and B, diagonal, of the form

$$\mathbf{A}=\left(\begin{array}{cccccc}{a}_{11}& a_{12}& 0& \dots & 0& 0\\ 0& a_{22}& a_{23}& \dots & 0& 0\\ 0& 0& a_{33}& \dots & 0& 0\\ ⋮& & & & & \\ 0& 0& 0& \dots & a_{N-1,N-1}& a_{N-1,N}\\ 0& 0& 0& \dots & a_{N,N-1}& a_{\mathit{NN}}\end{array}\right),$$E11

$$\mathbf{B}=\left(\begin{array}{cccccc}{b}_1& 0& 0& 0& \dots & 0\\ 0& b_2& 0& 0& \dots & 0\\ 0& 0& b_3& 0& \dots & 0\\ 0& 0& 0& b_4& \dots & 0\\ ⋮& & & & & \\ 0& 0& 0& 0& \dots & b_N\end{array}\right),$$E12

both matrices with complex entries.

A straightforward calculation yields the characteristic polynomial of the matrix A,

$$\begin{array}{l}det\left(\mathbf{A}-\lambda {\mathbf{I}}_N\right)=\left(a_{11}-\lambda \right)\cdots \left(a_{N-2,N-2}-\lambda \right)\\ \times \left[{\lambda }^2-\lambda \left(a_{\mathit{NN}}+a_{N-1,N-1}\right)-a_{N-1,N}{a}_{N,N-1}+a_{N-1,N-1}{a}_{\mathit{NN}}\right],\end{array}$$E13

whose solutions are

$$\lambda =a_{11},a_{22},\dots ,a_{N-2,N-2},a^{+},a^{-},$$E14

where

$$a^{\pm }=\frac{1}{2}\left(a_{N-1,N-1}+a_{\mathit{NN}}\pm \sqrt{{\left(a_{N-1,N-1}-a_{\mathit{NN}}\right)}^2+4a_{N-1,N}{a}_{N,N-1}}\right).$$E15

The corresponding eigenvectors are a bit complicated to write them down here, but, for instance, for dimension four, the eigenvalues are $a_{11},\left(a_{22}+a_{33}-z\right)/2,\left(a_{22}+a_{33}+z\right)/2$, and the corresponding eigenvectors are

$${\left(\mathrm{1,0,0}\right)}^T,$$E16

$${\left(\frac{a_{12}\left(-a_{22}+a_{33}+z\right)}{a_{32}\left(2a_{11}-a_{22}-a_{33}+z\right)},-\frac{-a_{22}+a_{33}+z}{2a_{32}},1\right)}^T,$$E17

$${\left(\frac{a_{12}\left(a_{22}-a_{33}+z\right)}{a_{32}\left(-2a_{11}+a_{22}+a_{33}+z\right)},\frac{a_{22}-a_{33}+z}{2a_{32}},1\right)}^T,$$E18

where

$$z=\sqrt{4a_{23}{a}_{32}+{\left(a_{22}-a_{33}\right)}^2}.$$E19

These eigenvalues are discrete and fixed and the eigenvectors are different between themselves.

Still, we can generate additional eigenvalues and eigenvectors by choosing the values of $a_{\mathit{jj}}$. The eigenvalue equation $\mathbf{Au}=\gamma \mathbf{u}$, $\gamma \in \mathbb{C}$, $\mathbf{u}={\left(u_1,\dots ,u_N\right)}^T\in {\mathbb{C}}^N$, leads to a system of simultaneous equations which is now solved for the $a_{\mathit{jj}}$’s with solutions

$$a_{11}=\gamma -\frac{a_{12}{u}_2}{u_1},\dots ,a_{N-1,N-1}=\gamma -\frac{a_{N-1,N}{u}_N}{u_{N-1}},a_{\mathit{NN}}=\gamma -\frac{a_{N,N-1}{u}_{N-1}}{u_N}.$$E20

This solution set gives rise to a matrix that results in weighted differences between the entries of a vector, resembling a finite-differences derivative [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

Now, the determinant of the matrix A is not zero; it is equal to

$$det\left(\mathbf{A}\right)=\left(a_{N-1,N-1}{a}_{\mathit{NN}}-a_{N-1,N}{a}_{N,N-1}\right)\prod _{j=1}^{N-2}{a}_{\mathit{jj}},$$E21

and then, we can compute its inverse in the usual way. We could find it with ${\mathbf{A}}^{-1}={\left[det\left(\mathbf{A}\right)\right]}^{-1}\mathrm{adj}\mathbf{A}$, where the adj A is the adjugate of A, whose entries are given by ${\left(-1\right)}^{i+j}det\left(\mathbf{A}\left(j,i\right)\right)$, where $\mathbf{A}\left(j,i\right)$ is the minor obtained by deleting the jth row and ith column from A [1]. The inverse matrix A becomes

$${\mathbf{A}}^{-1}=\left(\begin{array}{cccccc}\frac{1}{a_{11}}& -\frac{a_{12}}{a_{11}{a}_{22}}& \frac{a_{12}{a}_{23}}{a_{11}{a}_{22}{a}_{33}}& \dots & -\frac{a_{\mathit{NN}}{\prod }_{k=1}^{N-2}{a}_{k,k+1}}{w{\prod }_{k=i}^{N-2}{a}_{\mathit{kk}}}& \frac{{\prod }_{k=1}^{N-1}{a}_{k,k+1}}{w{\prod }_{k=1}^{N-2}{a}_{\mathit{kk}}}\\ 0& \frac{1}{a_{22}}& -\frac{a_{23}}{a_{22}{a}_{33}}& \dots & \frac{a_{\mathit{NN}}{\prod }_{k=2}^{N-2}{a}_{k,k+1}}{w{\prod }_{k=2}^{N-2}{a}_{\mathit{kk}}}& -\frac{{\prod }_{k=2}^{N-1}{a}_{k,k+1}}{w{\prod }_{k=2}^{N-2}{a}_{\mathit{kk}}}\\ ⋮& & & & & \\ 0& 0& 0& \dots & \frac{a_{\mathit{NN}}}{w}& -\frac{a_{N-1,N}}{w}\\ 0& 0& 0& \dots & -\frac{a_{N,N-1}}{w}& \frac{a_{N-1,N-1}}{w}\end{array}\right),$$E22

where $w=a_{N-1,N-1}{a}_{\mathit{NN}}-a_{N-1,N}{a}_{N,N-1}$. When w vanishes, there is no inverse of the matrix A. This matrix resembles a discrete integration approximation over subintervals, with weight factors given by the matrix elements [3].

The usual eigenvectors are too complicated to write them here, but the complement eigenvector $\mathbf{v}={\left(v_1,v_2,\dots ,v_N\right)}^T$, with eigenvalue $\tau$, is obtained when the diagonal matrix elements are given as

$$a_{\mathit{jj}}=\frac{1}{\tau }-\frac{v_{j+1}}{v_j}{a}_{j,j+1},1\le j<N,$$E23

$$a_{\mathit{NN}}=\frac{1}{\tau }-\frac{v_{N-1}}{v_N}{a}_{N,N-1}.$$E24

We saw some properties of the general matrix A. Next, we analyze particular versions of this matrix.

3. The commutator between matrices A and B

The commutator between the matrices A and B results in

$$\mathbf{A},\mathbf{B}=\left(\begin{array}{cccccc}0& a_{12}{\mathrm{\Delta }}_1& 0& \dots & 0& 0\\ 0& 0& a_{23}{\mathrm{\Delta }}_2& \dots & 0& 0\\ 0& 0& 0& \dots & 0& 0\\ ⋮& & & & & \\ 0& 0& 0& \dots & 0& a_{N-1,N}{\mathrm{\Delta }}_{N-1}\\ 0& 0& 0& \dots & -a_{N,N-1}{\mathrm{\Delta }}_{N-1}& 0\end{array}\right),$$E25

where ${\mathrm{\Delta }}_j=b_{j+1}-b_j$ is the difference between $b_{j+1}$ and $b_j$. This matrix is not diagonal; it does not depend on the diagonal elements of the matrix A, and it introduces the differences between adjacent diagonal elements of the matrix B. We can choose the values of $a_{j,j+1}$ and $a_{N,N-1}$ to obtain the effect of a diagonal commutator when acting on the particular arbitrary vector h. This procedure can be applied to more general matrices A and B. Still, our choice of the form of matrices A and B is for the sake of simplicity and to define a two-point derivative matrix that satisfies the commutator relationship and has as eigenvector the exponential function.

We ask matrices A and B to comply with the requirement that

$$\left[\mathbf{A},\mathbf{B}\right]\mathbf{h}=\alpha \mathbf{h},\alpha \in \mathbb{C},$$E26

where $\mathbf{h}={\left(h_1,h_2,\dots ,h_N\right)}^T\in {\mathbb{C}}^N$ is an arbitrary complex vector of length N but with non-zero entries $h_j\ne 0$. The requirement Eq. (26) leads to a system of simultaneous equations that are solved not for the vector h but for the entries of the matrixA. The resulting matrix is

$${\mathbf{A}}_c=\left(\begin{array}{cccccc}{a}_{11}& \frac{h_1\alpha }{h_2{\mathrm{\Delta }}_1}& 0& \dots & 0& 0\\ 0& a_{22}& \frac{h_2\alpha }{h_3{\mathrm{\Delta }}_2}& \dots & 0& 0\\ 0& 0& a_{33}& \dots & 0& 0\\ ⋮& & & & & \\ 0& 0& 0& \dots & a_{N-1,N-1}& \frac{h_{N-1}\alpha }{h_N{\mathrm{\Delta }}_{N-1}}\\ 0& 0& 0& \dots & -\frac{h_N\alpha }{h_{N-1}{\mathrm{\Delta }}_{N-1}}& a_{N,N}\end{array}\right),$$E27

which depends on h and $\alpha$. With these matrices, the commutator becomes a permutation matrix with weights $\alpha h_j/h_{j+1}$ or $\alpha h_N/h_{N-1}$,

$${\mathbf{A}}_c,\mathbf{B}=\left(\begin{array}{cccccc}0& \alpha \frac{h_1}{h_2}& 0& \dots & 0& 0\\ 0& 0& \alpha \frac{h_2}{h_3}& \dots & 0& 0\\ 0& 0& 0& \dots & 0& 0\\ ⋮& & & & & \\ 0& 0& 0& \dots & 0& \alpha \frac{h_{N-1}}{h_N}\\ 0& 0& 0& \dots & \alpha \frac{h_N}{h_{N-1}}& 0\end{array}\right).$$E28

This matrix performs two shifts when acting on the vector h. Since the matrix has only upper off-diagonal elements different from zero, there is a shift to the left. But the ratios $h_j/h_{j+1}$ shift back the vector. Then, the commutator is diagonal along the direction of h.

Now, we use another condition on ${\mathbf{A}}_c$ to determine its diagonal entries. The version of the matrix ${\mathbf{A}}_c$ that complies with the eigenvalue equation ${\tilde{\mathbf{A}}}_c\mathbf{v}=\gamma \mathbf{v}$ is

$$\begin{array}{l}\tilde{\mathbf{A}}=\\ \left(\begin{array}{cccccc}\gamma -\frac{h_1{v}_2\alpha }{h_2{v}_1{\mathrm{\Delta }}_1}& \frac{h_1\alpha }{h_2{\mathrm{\Delta }}_1}& 0& \dots & 0& 0\\ 0& \gamma -\frac{h_2{v}_3\alpha }{h_3{v}_2{\mathrm{\Delta }}_2}& \frac{h_2\alpha }{h_3{\mathrm{\Delta }}_2}& \dots & 0& 0\\ 0& 0& \gamma -\frac{h_3{v}_4\alpha }{h_4{v}_3{\mathrm{\Delta }}_3}& \dots & 0& 0\\ ⋮& & & & & \\ 0& 0& 0& \dots & \gamma -\frac{h_{N-1}{v}_N\alpha }{h_N{v}_{N-1}{\mathrm{\Delta }}_{N-1}}& \frac{h_{N-1}\alpha }{h_N{\mathrm{\Delta }}_{N-1}}\\ 0& 0& 0& \dots & -\frac{h_N\alpha }{h_{N-1}{\mathrm{\Delta }}_{N-1}}& \frac{h_N{v}_{N-1}\alpha }{h_{N-1}{v}_N{\mathrm{\Delta }}_{N-1}}+\gamma \end{array}\right).\end{array}$$E29

The action of this matrix on a vector $\mathbf{g}={\left(g_1,g_2,\dots ,g_N\right)}^T\in {\mathbb{C}}^N$ is

$${\left(\tilde{\mathbf{A}}\mathbf{g}\right)}_j=\frac{h_j\alpha }{h_{j+1}{\mathrm{\Delta }}_j}{g}_{j+1}+\left(\gamma -\frac{h_j{v}_{j+1}\alpha }{h_{j+1}{v}_j{\mathrm{\Delta }}_j}\right)g_j,1\le j<N,$$E30

$${\left(\tilde{\mathbf{A}}\mathbf{g}\right)}_N=\left(\gamma +\frac{h_N{v}_{N-1}\alpha }{h_{N-1}{v}_N{\mathrm{\Delta }}_{N-1}}\right)g_N-\frac{h_N\alpha }{h_{N-1}{\mathrm{\Delta }}_{N-1}}{g}_{N-1}.$$E31

If we use power series expansions for the quotients $h\left(b\right)/h\left(b+\mathrm{\Delta }\right)$ and $v\left(b+\mathrm{\Delta }\right)/v\left(b\right)$, at mesh points, we obtain

$$\frac{h_j}{h_{j+1}{\mathrm{\Delta }}_j}=\frac{1}{{\mathrm{\Delta }}_j}-\frac{h_j^{\prime }}{h_j}+O\left({\mathrm{\Delta }}_j\right),\text{and}\frac{v_{j+1}}{v_j{\mathrm{\Delta }}_j}=\frac{1}{{\mathrm{\Delta }}_j}+\frac{v_j^{\prime }}{v_j}+O\left({\mathrm{\Delta }}_j\right).$$E32

With these expansions, we obtain the small ${\mathrm{\Delta }}_j$ expressions

$${\left(\tilde{\mathbf{A}}\mathbf{g}\right)}_j\underset{{\mathrm{\Delta }}_j\to 0}{\to }\alpha \frac{g_{j+1}-g_j}{{\mathrm{\Delta }}_j}+g_j\left(\gamma -\alpha \frac{v_j^{\prime }}{v_j}\right)-\alpha \frac{h_j^{\prime }}{h_j}\left(g_{j+1}-g_j\right)+O\left({\mathrm{\Delta }}_j\right),1\le j<N,$$E33

$${\left(\tilde{\mathbf{A}}\mathbf{g}\right)}_N\underset{{\mathrm{\Delta }}_j\to 0}{\to }\alpha \frac{h_N}{h_{N-1}{\mathrm{\Delta }}_{N-1}}\left(\frac{v_{N-1}}{v_N}{g}_N-g_{N-1}\right)+\gamma g_N+O\left({\mathrm{\Delta }}_j\right).$$E34

Note that, since $g_{j+1}-g_j\to 0$ and $\left(g_{j+1}-g_j\right)/{\mathrm{\Delta }}_j\to g^{\prime }\left(b_j\right)$ as ${\mathrm{\Delta }}_j\to 0$, if we ask that $\gamma -\alpha v_j^{\prime }/v_j\to 0$, we would get $\alpha g^{\prime }\left(b_j\right)$ from Eqs. (32) and (33) in the limit ${\mathrm{\Delta }}_j\to 0$. Thus, if we make the choice $v\left(b\right)=e^{\mathit{\gamma b}/\alpha }$, Eqs. (30) and (31) lead to finite difference approximations to the derivative when A acts on g. The right hand side of Eq. (33) becomes the finite differences derivative of $g\left(b\right)$ at the boundary plus the boundary term $\gamma g_N$. Hereafter, we will use $v\left(b\right)=e^{\mathit{\gamma b}/\alpha }$, and finite ${\mathrm{\Delta }}_j$.

4. The derivative matrix

With the choices $h_j=v_j$ and $v_j=v\left(b_j\right)=e^{\gamma b_j/\alpha }$, the matrix ${\alpha }^{-1}\tilde{\mathbf{A}}$ becomes the derivative matrix

$$\mathbf{D}=\left(\begin{array}{cccccc}\frac{\gamma }{\alpha }-\frac{1}{{\mathrm{\Delta }}_1}& \frac{e^{-\frac{\gamma {\mathrm{\Delta }}_1}{\alpha }}}{{\mathrm{\Delta }}_1}& 0& \dots & 0& 0\\ 0& \frac{\gamma }{\alpha }-\frac{1}{{\mathrm{\Delta }}_2}& \frac{e^{-\frac{\gamma {\mathrm{\Delta }}_2}{\alpha }}}{{\mathrm{\Delta }}_2}& \dots & 0& 0\\ 0& 0& \frac{\gamma }{\alpha }-\frac{1}{{\mathrm{\Delta }}_3}& \dots & 0& 0\\ ⋮& & & & & \\ 0& 0& 0& \dots & \frac{e^{-\frac{\gamma {\mathrm{\Delta }}_{N-2}}{\alpha }}}{{\mathrm{\Delta }}_{N-2}}& 0\\ 0& 0& 0& \dots & \frac{\gamma }{\alpha }-\frac{1}{{\mathrm{\Delta }}_{N-1}}& \frac{e^{-\frac{\gamma {\mathrm{\Delta }}_{N-1}}{\alpha }}}{{\mathrm{\Delta }}_{N-1}}\\ 0& 0& 0& \dots & -\frac{e^{\frac{\gamma {\mathrm{\Delta }}_{N-1}}{\alpha }}}{{\mathrm{\Delta }}_{N-1}}& \frac{\gamma }{\alpha }+\frac{1}{{\mathrm{\Delta }}_{N-1}}\end{array}\right).$$E35

When the matrix D acts to the left on a vector ${\mathbf{f}}^T={\left(f_1,f_2,\dots ,f_N\right)}^T\in {\mathbb{C}}^N$, we obtain

$${\mathbf{f}}^T\mathbf{D}=-{\left({\left(\mathcal{B}\mathbf{f}\right)}_1,{\left(\mathcal{D}\mathbf{f}\right)}_2,\dots ,{\left(\mathcal{D}\mathbf{f}\right)}_{N-2},{\left(\mathcal{D}\mathbf{f}\right)}_{N-1}+{\left(\mathcal{B}\mathbf{f}\right)}_{N-1},,{\left(\mathcal{D}\mathbf{f}\right)}_N+{\left(\mathcal{B}\mathbf{f}\right)}_N\right)}^T,$$E36

where

$${\left(\mathcal{D}\mathbf{f}\right)}_j=\left(\frac{\gamma }{\alpha }-\frac{1}{{\mathrm{\Delta }}_j}\right)f_j+e^{-\gamma {\mathrm{\Delta }}_{j-1}/\alpha }\frac{f_{j-1}}{{\mathrm{\Delta }}_{j-1}},1<j\le N,$$E37

$${\left(\mathcal{B}\mathbf{f}\right)}_1=\left(\frac{1}{{\mathrm{\Delta }}_1}-\frac{\gamma }{\alpha }\right)f_1,$$E38

$${\left(\mathcal{B}\mathbf{f}\right)}_{N-1}=e^{\frac{\gamma }{\alpha }{\mathrm{\Delta }}_{N-1}}\frac{f_N}{{\mathrm{\Delta }}_{N-1}},$$E39

$${\left(\mathcal{B}\mathbf{f}\right)}_N=-2\frac{f_N}{{\mathrm{\Delta }}_{N-1}},$$E40

which is the negative of a finite-differences approximation to the derivative of f. When ${\mathrm{\Delta }}_j$ is small and using the exponential expansions $e^{\mp \gamma {\mathrm{\Delta }}_j/\alpha }=1\mp \gamma \frac{{\mathrm{\Delta }}_j}{\alpha }+\cdots$, we find that

$${\left(\mathcal{D}\mathbf{f}\right)}_j\underset{{\mathrm{\Delta }}_j\to 0}{\to }\frac{f_j}{{\mathrm{\Delta }}_j}-\frac{f_{j-1}}{{\mathrm{\Delta }}_{j-1}}-\frac{\gamma }{\alpha }\left(f_j-f_{j-1}\right)+O\left({\mathrm{\Delta }}_j\right),1<j<N.$$E41

When D is applied to the right to the vector $\mathbf{g}={\left(g_1,\dots ,g_N\right)}^T$, we obtain

$$\mathbf{Dg}={\left({\left(\mathrm{D}\mathbf{g}\right)}_1,{\left(\mathrm{D}\mathbf{g}\right)}_2,\dots ,{\left(\mathrm{D}\mathbf{g}\right)}_{N-1},{\left(\mathrm{D}\mathbf{g}\right)}_N\right)}^T,$$E42

where

$${\left(\mathrm{D}\mathbf{g}\right)}_j=\left(\frac{\gamma }{\alpha }-\frac{1}{{\mathrm{\Delta }}_j}\right)g_j+e^{-\frac{\gamma }{\alpha }{\mathrm{\Delta }}_j}\frac{g_{j+1}}{{\mathrm{\Delta }}_j},1\le j\le N-1,$$E43

$${\left(\mathrm{D}\mathbf{g}\right)}_N=\left(\frac{\gamma }{\alpha }+\frac{1}{{\mathrm{\Delta }}_{N-1}}\right)g_N-e^{\frac{\gamma }{\alpha }{\mathrm{\Delta }}_{N-1}}\frac{g_{N-1}}{{\mathrm{\Delta }}_{N-1}}.$$E44

For small ${\mathrm{\Delta }}_j$, these equalities become

$${\left(\mathrm{D}\mathbf{g}\right)}_j\underset{{\mathrm{\Delta }}_j\to 0}{\to }\frac{g_{j+1}-g_j}{{\mathrm{\Delta }}_j}-\frac{\gamma }{\alpha }\left(g_{j+1}-g_j\right)+O\left({\mathrm{\Delta }}_j\right),1\le j\le N-1,$$E45

$${\left(\mathrm{D}\mathbf{g}\right)}_N\underset{{\mathrm{\Delta }}_j\to 0}{\to }\frac{g_N-g_{N-1}}{{\mathrm{\Delta }}_{N-1}}+\frac{\gamma }{\alpha }\left(g_N-g_{N-1}\right)+O\left({\mathrm{\Delta }}_j\right).$$E46

These approximations show that D is a derivation matrix, exact for the exponential function $e^{\mathit{\gamma b}/\alpha }$ for any value of ${\mathrm{\Delta }}_j$.

For instance, for dimension five, the eigenvalues of the derivative matrix D are $\left\{\frac{\gamma }{\alpha },\frac{\gamma }{\alpha },\frac{\gamma }{\alpha }-\frac{1}{{\mathrm{\Delta }}_1},\frac{\gamma }{\alpha }-\frac{1}{{\mathrm{\Delta }}_2},\frac{\gamma }{\alpha }-\frac{1}{{\mathrm{\Delta }}_{N-2}}\right\}$, and the corresponding eigenvectors are

$${\left(e^{\frac{\gamma }{\alpha }{b}_1},e^{\frac{\gamma }{\alpha }{b}_2},\dots ,e^{\frac{\gamma }{\alpha }{b}_4},e^{\frac{\gamma }{\alpha }{b}_5}\right)}^T,$$E47

$${\left(\mathrm{0,0,0,0,0}\right)}^T,$$E48

$${\left(\mathrm{1,0,0,0,0}\right)}^T,$$E49

$${\left(-\frac{e^{\frac{\gamma }{\alpha }{b}_1}{\mathrm{\Delta }}_2}{{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2},e^{\frac{\gamma }{\alpha }{b}_2}\mathrm{,0,0,0}\right)}^T,$$E50

$${\left(-\frac{e^{\frac{\gamma }{\alpha }{b}_1}{\mathrm{\Delta }}_3^2}{\left({\mathrm{\Delta }}_1-{\mathrm{\Delta }}_3\right)\left(-{\mathrm{\Delta }}_2+{\mathrm{\Delta }}_3\right)},-\frac{e^{\frac{\gamma }{\alpha }{b}_2}{\mathrm{\Delta }}_3}{{\mathrm{\Delta }}_2-{\mathrm{\Delta }}_3},e^{\frac{\gamma }{\alpha }{b}_3}\mathrm{,0,0}\right)}^T,$$E51

and the exponential function is an eigenfunction of the derivative matrix with eigenvalue $\gamma /\alpha$.

The determinant of the derivative matrix D is

$$\mid \mathbf{D}\mid =\frac{{\gamma }^2}{{\alpha }^2}\prod _{k=1}^{N-2}\left(\frac{1}{{\mathrm{\Delta }}_k}-\frac{\gamma }{\alpha }\right).$$E52

Therefore, as long as $\alpha ,{\mathrm{\Delta }}_k,\gamma \ne 0$, and $\alpha -\gamma {\mathrm{\Delta }}_k\ne 0$, $k=1,2,\dots ,N-2$, we can compute the inverse of D. For dimension $4\times 4$, the inverse matrix is

$$\begin{array}{l}{\mathbf{D}}^{-1}=\\ \left(\begin{array}{cccc}-\frac{\alpha {\mathrm{\Delta }}_1}{\alpha -\gamma {\mathrm{\Delta }}_1}& -\frac{e^{-\frac{\gamma }{\alpha }{\mathrm{\Delta }}_1}{\alpha }^2{\mathrm{\Delta }}_2}{\left(\alpha -\gamma {\mathrm{\Delta }}_1\right)\left(\alpha -\gamma {\mathrm{\Delta }}_2\right)}& \frac{e^{-\frac{\gamma }{\alpha }\left({\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2\right)}{\alpha }^3\left(\alpha +\gamma {\mathrm{\Delta }}_4\right)}{{\gamma }^2\left(\alpha -\gamma {\mathrm{\Delta }}_1\right)\left(\alpha -\gamma {\mathrm{\Delta }}_2\right){\mathrm{\Delta }}_4}& -\frac{e^{-\frac{\gamma }{\alpha }\left({\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2+{\mathrm{\Delta }}_3\right)}{\alpha }^4}{{\gamma }^2\left(\alpha -\gamma {\mathrm{\Delta }}_1\right)\left(\alpha -\gamma {\mathrm{\Delta }}_2\right){\mathrm{\Delta }}_4}\\ 0& -\frac{e^{-\frac{\gamma }{\alpha }{\mathrm{\Delta }}_2}\alpha {\mathrm{\Delta }}_2}{\alpha -\gamma {\mathrm{\Delta }}_2}& \frac{e^{-\frac{\gamma }{\alpha }\left({\mathrm{\Delta }}_2+{\mathrm{\Delta }}_3\right)}{\alpha }^2\left(\alpha +\gamma {\mathrm{\Delta }}_4\right)}{{\gamma }^2\left(\alpha -\gamma {\mathrm{\Delta }}_2\right){\mathrm{\Delta }}_4}& -\frac{e^{-\frac{\gamma \left({\mathrm{\Delta }}_2+{\mathrm{\Delta }}_3\right)}{\alpha }}{\alpha }^3}{{\gamma }^2\left(\alpha -\gamma {\mathrm{\Delta }}_2\right){\mathrm{\Delta }}_4}\\ 0& 0& \frac{\alpha \left(\alpha +\gamma {\mathrm{\Delta }}_4\right)}{{\gamma }^2{\mathrm{\Delta }}_4}& -\frac{e^{-\frac{\gamma {\mathrm{\Delta }}_3}{\alpha }}{\alpha }^2}{{\gamma }^2{\mathrm{\Delta }}_4}\\ 0& 0& \frac{e^{\frac{\gamma {\mathrm{\Delta }}_3}{\alpha }}{\alpha }^2}{{\gamma }^2{\mathrm{\Delta }}_3}& -\frac{\alpha \left(\alpha -\gamma {\mathrm{\Delta }}_3\right)}{{\gamma }^2{\mathrm{\Delta }}_3}\end{array}\right)\end{array}$$E53

When this matrix acts on a vector, the result is a collection of partial summations with weights given by its entries, an integration matrix. The eigenvalues of the inverse matrix are $\frac{\alpha }{\gamma },\frac{\alpha }{\gamma },-\frac{\alpha {\mathrm{\Delta }}_1}{\alpha -\gamma {\mathrm{\Delta }}_1},-\frac{\alpha {\mathrm{\Delta }}_2}{\alpha -\gamma {\mathrm{\Delta }}_2}$ and the corresponding eigenvectors are

$${\left(e^{\frac{\gamma }{\alpha }{b}_1},e^{\frac{\gamma }{\alpha }{b}_2},e^{\frac{\gamma }{\alpha }{b}_3},e^{\frac{\gamma }{\alpha }{b}_4}\right)}^T,$$E54

$${\left(\mathrm{0,0,0,0}\right)}^T,$$E55

$${\left(e^{\frac{\gamma }{\alpha }{b}_1}\mathrm{,0,0,0}\right)}^T,$$E56

$${\left(-\frac{e^{\frac{\gamma }{\alpha }{b}_1}{\mathrm{\Delta }}_2}{{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2},e^{\frac{\gamma }{\alpha }{b}_2}\mathrm{,0,0}\right)}^T.$$E57

The exponential function $e^{\mathit{\gamma b}/\alpha }$ is an eigenvector of the inverse matrix (a finite differences integration matrix) with eigenvalue $\alpha /\gamma$.

4.1 Summation by parts

A practical result is the summation by parts theorem (the discrete version of the continuous variable integration by parts theorem), the subject of this section.

We start by defining the summation matrix

$$\mathbf{S}=\left(\begin{array}{cccccc}{\mathrm{\Delta }}_1& 0& 0& \dots & 0& 0\\ 0& {\mathrm{\Delta }}_2& 0& \dots & 0& 0\\ ⋮& & & & & \\ 0& 0& 0& \dots & {\mathrm{\Delta }}_{N-1}& 0\\ 0& 0& 0& \dots & 0& 0\end{array}\right).$$E58

The summation matrix S allows for determining the equality

$${\mathbf{f}}^T\mathbf{SDg}={\mathbf{g}}^T\left(\tilde{\mathbf{B}}-{\mathbf{SD}}^{†}\right)\mathbf{f},$$E59

where

$$\tilde{\mathbf{B}}=\left(\begin{array}{cccccc}\frac{\gamma }{\alpha }{\mathrm{\Delta }}_1-1& 0& \dots & 0& 0& 0\\ 0& 0& 0& 0& 0& \\ 0& 0& \dots & 0& 0& 0\\ ⋮& & & & & \\ 0& 0& \dots & 0& 0& 0\\ 0& 0& \dots & 0& e^{-\frac{\gamma {\mathrm{\Delta }}_{N-1}}{\alpha }}& 0\end{array}\right)$$E60

is the boundary matrix, and

$${\mathbf{D}}^{†}=\left(\begin{array}{ccccccc}0& 0& 0& \dots & 0& 0& 0\\ -\frac{e^{-\frac{\gamma {\mathrm{\Delta }}_1}{\alpha }}}{{\mathrm{\Delta }}_2}& \frac{1}{{\mathrm{\Delta }}_2}-\frac{\gamma }{\alpha }& 0& \dots & 0& 0& 0\\ 0& -\frac{e^{-\frac{\gamma {\mathrm{\Delta }}_2}{\alpha }}}{{\mathrm{\Delta }}_3}& \frac{1}{{\mathrm{\Delta }}_3}-\frac{\gamma }{\alpha }& \dots & 0& 0& 0\\ ⋮& & & & & & \\ 0& 0& 0& 0& \dots & -\frac{e^{-\frac{\gamma {\mathrm{\Delta }}_{N-2}}{\alpha }}}{{\mathrm{\Delta }}_{N-1}}& \frac{1}{{\mathrm{\Delta }}_{N-1}}-\frac{\gamma }{\alpha }\\ 0& 0& 0& \dots & 0& 0& 0\end{array}\right)$$E61

is the with continuous entries, adjoint of the discrete derivative matrix D. When this matrix acts on a vector f, we obtain a vector with entries

$${\left({\mathbf{D}}^{†}\mathbf{f}\right)}_j=\left(\frac{1}{{\mathrm{\Delta }}_j}-\frac{\gamma }{\alpha }\right)f_j-\frac{1}{{\mathrm{\Delta }}_j}{e}^{-\frac{\gamma {\mathrm{\Delta }}_{j-1}}{\alpha }}{f}_{j-1},1<j<N.$$E62

The small ${\mathrm{\Delta }}_j$ limit of the boundary terms is

$${\left(\mathbf{g}\tilde{\mathbf{B}}\mathbf{f}\right)}_1\underset{{\mathrm{\Delta }}_j\to 0}{\to }-f_1{g}_1+O\left({\mathrm{\Delta }}_j\right),$$E63

$${\left(\mathbf{g}\tilde{\mathbf{B}}\mathbf{f}\right)}_N\underset{{\mathrm{\Delta }}_j\to 0}{\to }{f}_{N-1}{g}_N+O\left({\mathrm{\Delta }}_j\right),$$E64

and, for the adjoint matrix ${\mathbf{D}}^{†}$, we obtain

$${\left({\mathbf{D}}^{†}\mathbf{f}\right)}_j\underset{{\mathrm{\Delta }}_j\to 0}{\to }\frac{f_j-f_{j-1}}{{\mathrm{\Delta }}_j}-\frac{\gamma }{\alpha }{f}_j+O\left({\mathrm{\Delta }}_j\right).$$E65

Thus, the adjoint matrix is a derivation minus a constant term.

For a matrix with dimension $4\times 4$, for instance, the eigenvalues are 0 of multiplicity two, $\frac{1}{{\mathrm{\Delta }}_2}-\frac{\gamma }{\alpha }$, and $\frac{1}{{\mathrm{\Delta }}_3}-\frac{\gamma }{\alpha }$, and the eigenvectors are

$${\left(\mathrm{0,0,0,1}\right)}^T,$$E66

$${\left(e^{-\frac{b_1\gamma }{\alpha }}\left(1-{\mathrm{\Delta }}_2\frac{\gamma }{\alpha }\right)\left(1-{\mathrm{\Delta }}_3\frac{\gamma }{\alpha }\right),e^{-\frac{b_2\gamma }{\alpha }}\left(1-{\mathrm{\Delta }}_3\frac{\gamma }{\alpha }\right),e^{-\frac{b_3\gamma }{\alpha }},0\right)}^T,$$E67

$${\left(0,-\frac{\left({\mathrm{\Delta }}_3-{\mathrm{\Delta }}_2\right)e^{-\frac{b_2\gamma }{\alpha }}}{{\mathrm{\Delta }}_2},e^{-\frac{b_3\gamma }{\alpha }},0\right)}^T,$$E68

$${\left(\mathrm{0,0,1,0}\right)}^T,$$E69

and the complement, for arbitrary dimension, eigenvector is

$${\left(0,\frac{e^{-\gamma b_2/\alpha }}{\alpha -\left(\mathit{\alpha \tau }+\gamma \right){\mathrm{\Delta }}_2},\dots ,\frac{e^{-\gamma b_{N-1}/\alpha }}{{\prod }_{k=2}^{N-2}\left(\alpha -\left(\mathit{\alpha \tau }+\gamma \right){\mathrm{\Delta }}_k\right)},0\right)}^T,$$E70

with eigenvalue $\tau \in \mathbb{C}$, and $\tau \cancel{=}\frac{1}{{\mathrm{\Delta }}_k}-\frac{\gamma }{\alpha }$, $k=2,3,\dots ,N-1$, the complement of the regular eigenvalues.

The determinant of the matrix ${\mathbf{D}}^{†}$ vanishes, and then, there is no inverse matrix at all.

We arrived at a finite-differences derivative of a function that complies with the discrete versions of the properties that the continuous derivative has [16].

4.2 The upper diagonal matrix

The simplest case of matrices with diagonal commutator along the direction of h is obtained when the diagonal elements of the matrix A vanish, $a_{\mathit{jj}}=0$. That matrix is

$${\mathbf{A}}_2=\left(\begin{array}{cccccc}0& \frac{h_1\alpha }{h_2{\mathrm{\Delta }}_1}& 0& \dots & 0& 0\\ 0& 0& \frac{h_2\alpha }{h_3{\mathrm{\Delta }}_2}& \dots & 0& 0\\ 0& 0& 0& \dots & 0& 0\\ ⋮& & & & & \\ 0& 0& 0& \dots & 0& \frac{h_{N-1}\alpha }{h_N{\mathrm{\Delta }}_{N-1}}\\ 0& 0& 0& \dots & -\frac{h_N\alpha }{h_{N-1}{\mathrm{\Delta }}_{N-1}}& 0\end{array}\right).$$E71

This matrix is also a cyclic shifting matrix to the left, with rescaling, in general, and only a rescaling matrix when acting on the vector h as if it were the matrix $\alpha diag\left({\mathrm{\Delta }}_j^{-1}\right)$, with ${\mathrm{\Delta }}_N={\mathrm{\Delta }}_{N-1}$.

The eigenvalues of the matrix ${\mathbf{A}}_2$ are

$$\lambda =0\left(\text{with multiplicity}N-2\right),\mathit{\text{and}}{\lambda }_{\pm }=\pm \frac{i\alpha }{{\mathrm{\Delta }}_{N-1}},$$E72

and the corresponding eigenvectors are

$${\mathbf{x}}_{\lambda }=\left(\begin{array}{c}1\\ 0\\ 0\\ ⋮\\ 0\end{array}\right),\left(\begin{array}{c}\frac{{\mathrm{\Delta }}_{N-1}^{N-2}{h}_1}{{\prod }_{j=1}^{N-2}{\mathrm{\Delta }}_j}\\ i\frac{{\mathrm{\Delta }}_{N-1}^{N-3}{h}_2}{{\prod }_{j=2}^{N-2}{\mathrm{\Delta }}_j}\\ -\frac{{\mathrm{\Delta }}_{N-1}^{N-4}{h}_3}{{\prod }_{j=3}^{N-2}{\mathrm{\Delta }}_j}\\ ⋮\\ i^{N-2}{h}_{N-1}\\ i^{N-1}{h}_N\end{array}\right),\left(\begin{array}{c}-\frac{{\mathrm{\Delta }}_{N-1}^{N-2}{h}_1}{{\prod }_{j=1}^{N-2}{\mathrm{\Delta }}_j}\\ -i\frac{{\mathrm{\Delta }}_{N-1}^{N-3}{h}_2}{{\prod }_{j=2}^{N-2}{\mathrm{\Delta }}_j}\\ \frac{{\mathrm{\Delta }}_{N-1}^{N-4}{h}_3}{{\prod }_{j=3}^{N-2}{\mathrm{\Delta }}_j}\\ ⋮\\ -i^{N-2}{h}_{N-1}\\ -i^{N-1}{h}_N\end{array}\right).$$E73

Additionally, there are $N-3$, not well-defined vectors corresponding to the null eigenvalue with degeneracy $N-3$. These eigenvectors give rise to a vector space of dimension three. However, if we just look for the solution of the simultaneous linear equations, there is another set of eigenvectors, the complement eigenvalue $\beta$, and the eigenvector v with entries—for dimension five—given by

$$\frac{v_5}{v_4}=-i\frac{h_5}{h_4},\frac{v_4}{v_3}=\frac{h_4\beta {\mathrm{\Delta }}_3}{h_3\alpha },\frac{v_3}{v_2}=\frac{h_3\beta {\mathrm{\Delta }}_2}{h_2},\frac{v_2}{v_1}=\frac{h_2\beta {\mathrm{\Delta }}_1}{h_1\alpha },v_1\in \mathbb{C}.$$E74

with eigenvalue $\beta =\pm i\alpha /{\mathrm{\Delta }}_4$.

The determinant of the matrix ${\mathbf{A}}_2$ vanishes, and then, there is no inverse matrix at all.

5. Conclusion

We discussed several properties of matrices, namely, pairs of matrices with a diagonal commutator when applied to a given vector, exact finite-differences derivation and integration, and complement eigenvalues and eigenvectors.

These results are relevant in quantum mechanics theory, in which some operators have a discrete spectrum. Our scheme might be of interest in quantum gravity theory, too, because the space is quantized, and then a discrete derivative with respect to the length variable is needed [17, 18].

Acknowledgments

A. Martínez-Pérez would like to acknowledge the support from the UNAM Postdoctoral Program (POSDOC).