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

R1=010…00001…00000…00⋮000…01100…00.E1

Given an arbitrary vector h=h1h2…hNT∈C, hj≠0, the action of R1 on this vector results in

R1h=h2h3…hNh1T.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

R2=h2h1000…00h3h200…000h4h30…0000h5h4…0⋮0000…h1hNE3

on h is R2h=h2h3…hNh1T. 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:

R3=0h1h200…0000h2h30…00000h3h4…00⋮0000…0hN−1hNhNh1000…00.E4

We have that R3h=h. The eigenvalues of this matrix are the same as those of R1, the N roots of unity: λn=ei2πn/N, n=0,1,2,…,N−1, and h is the eigenvector that corresponds to the eigenvalue λ0=1.

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

Q1=q11q1200q22q230q32q33E5

has eigenvalues q11,12q22+q33−z,12q22+q33+z, and the corresponding eigenvectors are

1,0,0T,E6

q12−q22+q33+zq322q11−q22−q33+z−−q22+q33+z2q321T,E7

q12q22−q33+zq32−2q11+q22+q33+zq22−q33+z2q321T,E8

where, z=4q23q32+q22−q332. These are the only eigenvalues and eigenvectors. The eigenvalues are fixed quantities and depend on the fixed entries of the matrix. However, the matrix

v1τv1−v2−v1τv1−v200v2τv2−v3−v2τv2−v30−v3τv3−v2v3τv3−v2,E9

has eigenvalues 0, τ, v1τv1−v2 with corresponding eigenvectors

1,1,1T,v1v2v3T,1,0,0T,E10

where the eigenvalue τ is independent of vj, 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∈ℕ, matrices A, with non-zero elements around and on the diagonal, and B, diagonal, of the form

A=a11a120…000a22a23…0000a33…00⋮000…aN−1,N−1aN−1,N000…aN,N−1aNN,E11

B=b1000…00b200…000b30…0000b4…0⋮0000…bN,E12

both matrices with complex entries.

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

detA−λIN=a11−λ⋯aN−2,N−2−λ×λ2−λaNN+aN−1,N−1−aN−1,NaN,N−1+aN−1,N−1aNN,E13

whose solutions are

λ=a11,a22,…,aN−2,N−2,a+,a−,E14

where

a±=12aN−1,N−1+aNN±aN−1,N−1−aNN2+4aN−1,NaN,N−1.E15

The corresponding eigenvectors are a bit complicated to write them down here, but, for instance, for dimension four, the eigenvalues are a11,a22+a33−z/2,a22+a33+z/2, and the corresponding eigenvectors are

1,0,0T,E16

a12−a22+a33+za322a11−a22−a33+z−−a22+a33+z2a321T,E17

a12a22−a33+za32−2a11+a22+a33+za22−a33+z2a321T,E18

where

z=4a23a32+a22−a332.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 ajj. The eigenvalue equation Au=γu, γ∈C, u=u1…uNT∈CN, leads to a system of simultaneous equations which is now solved for the ajj’s with solutions

a11=γ−a12u2u1,…,aN−1,N−1=γ−aN−1,NuNuN−1,aNN=γ−aN,N−1uN−1uN.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

detA=aN−1,N−1aNN−aN−1,NaN,N−1∏j=1N−2ajj,E21

and then, we can compute its inverse in the usual way. We could find it with A−1=detA−1adjA, where the adj A is the adjugate of A, whose entries are given by −1i+jdetAji, where Aji is the minor obtained by deleting the jth row and ith column from A [1]. The inverse matrix A becomes

A−1=1a11−a12a11a22a12a23a11a22a33…−aNN∏k=1N−2ak,k+1w∏k=iN−2akk∏k=1N−1ak,k+1w∏k=1N−2akk01a22−a23a22a33…aNN∏k=2N−2ak,k+1w∏k=2N−2akk−∏k=2N−1ak,k+1w∏k=2N−2akk⋮000…aNNw−aN−1,Nw000…−aN,N−1waN−1,N−1w,E22

where w=aN−1,N−1aNN−aN−1,NaN,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 v=v1v2…vNT, with eigenvalue τ, is obtained when the diagonal matrix elements are given as

ajj=1τ−vj+1vjaj,j+1,1≤j<N,E23

aNN=1τ−vN−1vNaN,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

A,B=0a12Δ10…0000a23Δ2…00000…00⋮000…0aN−1,NΔN−1000…−aN,N−1ΔN−10,E25

where Δj=bj+1−bj is the difference between bj+1 and bj. 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 aj,j+1 and aN,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

ABh=αh,α∈C,E26

where h=h1h2…hNT∈CN is an arbitrary complex vector of length N but with non-zero entries hj≠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

Ac=a11h1αh2Δ10…000a22h2αh3Δ2…0000a33…00⋮000…aN−1,N−1hN−1αhNΔN−1000…−hNαhN−1ΔN−1aN,N,E27

which depends on h and α. With these matrices, the commutator becomes a permutation matrix with weights αhj/hj+1 or αhN/hN−1,

Ac,B=0αh1h20…0000αh2h3…00000…00⋮000…0αhN−1hN000…αhNhN−10.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 hj/hj+1 shift back the vector. Then, the commutator is diagonal along the direction of h.

Now, we use another condition on Ac to determine its diagonal entries. The version of the matrix Ac that complies with the eigenvalue equation A˜cv=γv is

A˜=γ−h1v2αh2v1Δ1h1αh2Δ10…000γ−h2v3αh3v2Δ2h2αh3Δ2…0000γ−h3v4αh4v3Δ3…00⋮000…γ−hN−1vNαhNvN−1ΔN−1hN−1αhNΔN−1000…−hNαhN−1ΔN−1hNvN−1αhN−1vNΔN−1+γ.E29

The action of this matrix on a vector g=g1g2…gNT∈CN is

A˜gj=hjαhj+1Δjgj+1+γ−hjvj+1αhj+1vjΔjgj,1≤j<N,E30

A˜gN=γ+hNvN−1αhN−1vNΔN−1gN−hNαhN−1ΔN−1gN−1.E31

If we use power series expansions for the quotients hb/hb+Δ and vb+Δ/vb, at mesh points, we obtain

hjhj+1Δj=1Δj−hj′hj+OΔj,andvj+1vjΔj=1Δj+vj′vj+OΔj.E32

With these expansions, we obtain the small Δj expressions

A˜gj→Δj→0αgj+1−gjΔj+gjγ−αvj′vj−αhj′hjgj+1−gj+OΔj,1≤j<N,E33

A˜gN→Δj→0αhNhN−1ΔN−1vN−1vNgN−gN−1+γgN+OΔj.E34

Note that, since gj+1−gj→0 and gj+1−gj/Δj→g′bj as Δj→0, if we ask that γ−αvj′/vj→0, we would get αg′bj from Eqs. (32) and (33) in the limit Δj→0. Thus, if we make the choice vb=eγb/α, 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 gb at the boundary plus the boundary term γgN. Hereafter, we will use vb=eγb/α, and finite Δj.

4. The derivative matrix

With the choices hj=vj and vj=vbj=eγbj/α, the matrix α−1A˜ becomes the derivative matrix

D=γα−1Δ1e−γΔ1αΔ10…000γα−1Δ2e−γΔ2αΔ2…0000γα−1Δ3…00⋮000…e−γΔN−2αΔN−20000…γα−1ΔN−1e−γΔN−1αΔN−1000…−eγΔN−1αΔN−1γα+1ΔN−1.E35

When the matrix D acts to the left on a vector fT=f1f2…fNT∈CN, we obtain

fTD=−Bf1Df2…DfN−2DfN−1+BfN−1DfN+BfNT,E36

where

Dfj=γα−1Δjfj+e−γΔj−1/αfj−1Δj−1,1<j≤N,E37

Bf1=1Δ1−γαf1,E38

BfN−1=eγαΔN−1fNΔN−1,E39

BfN=−2fNΔN−1,E40

which is the negative of a finite-differences approximation to the derivative of f. When Δj is small and using the exponential expansions e∓γΔj/α=1∓γΔjα+⋯, we find that

Dfj→Δj→0fjΔj−fj−1Δj−1−γαfj−fj−1+OΔj,1<j<N.E41

When D is applied to the right to the vector g=g1…gNT, we obtain

Dg=Dg1Dg2…DgN−1DgNT,E42

where

Dgj=γα−1Δjgj+e−γαΔjgj+1Δj,1≤j≤N−1,E43

DgN=γα+1ΔN−1gN−eγαΔN−1gN−1ΔN−1.E44

For small Δj, these equalities become

Dgj→Δj→0gj+1−gjΔj−γαgj+1−gj+OΔj,1≤j≤N−1,E45

DgN→Δj→0gN−gN−1ΔN−1+γαgN−gN−1+OΔj.E46

These approximations show that D is a derivation matrix, exact for the exponential function eγb/α for any value of Δj.

For instance, for dimension five, the eigenvalues of the derivative matrix D are γαγαγα−1Δ1γα−1Δ2γα−1ΔN−2, and the corresponding eigenvectors are

eγαb1eγαb2…eγαb4eγαb5T,E47

0,0,0,0,0T,E48

1,0,0,0,0T,E49

−eγαb1Δ2Δ1−Δ2eγαb2,0,0,0T,E50

−eγαb1Δ32Δ1−Δ3−Δ2+Δ3−eγαb2Δ3Δ2−Δ3eγαb3,0,0T,E51

and the exponential function is an eigenfunction of the derivative matrix with eigenvalue γ/α.

The determinant of the derivative matrix D is

∣D∣=γ2α2∏k=1N−21Δk−γα.E52

Therefore, as long as α,Δk,γ≠0, and α−γΔk≠0, k=1,2,…,N−2, we can compute the inverse of D. For dimension 4×4, the inverse matrix is

D−1=−αΔ1α−γΔ1−e−γαΔ1α2Δ2α−γΔ1α−γΔ2e−γαΔ1+Δ2α3α+γΔ4γ2α−γΔ1α−γΔ2Δ4−e−γαΔ1+Δ2+Δ3α4γ2α−γΔ1α−γΔ2Δ40−e−γαΔ2αΔ2α−γΔ2e−γαΔ2+Δ3α2α+γΔ4γ2α−γΔ2Δ4−e−γΔ2+Δ3αα3γ2α−γΔ2Δ400αα+γΔ4γ2Δ4−e−γΔ3αα2γ2Δ400eγΔ3αα2γ2Δ3−αα−γΔ3γ2Δ3E53

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 αγ,αγ,−αΔ1α−γΔ1,−αΔ2α−γΔ2 and the corresponding eigenvectors are

eγαb1eγαb2eγαb3eγαb4T,E54

0,0,0,0T,E55

eγαb1,0,0,0T,E56

−eγαb1Δ2Δ1−Δ2eγαb2,0,0T.E57

The exponential function eγb/α is an eigenvector of the inverse matrix (a finite differences integration matrix) with eigenvalue α/γ.

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

S=Δ100…000Δ20…00⋮000…ΔN−10000…00.E58

The summation matrix S allows for determining the equality

fTSDg=gTB˜−SD†f,E59

where

B˜=γαΔ1−10…0000000000…000⋮00…00000…0e−γΔN−1α0E60

is the boundary matrix, and

D†=000…000−e−γΔ1αΔ21Δ2−γα0…0000−e−γΔ2αΔ31Δ3−γα…000⋮0000…−e−γΔN−2αΔN−11ΔN−1−γα000…000E61

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

D†fj=1Δj−γαfj−1Δje−γΔj−1αfj−1,1<j<N.E62

The small Δj limit of the boundary terms is

gB˜f1→Δj→0−f1g1+OΔj,E63

gB˜fN→Δj→0fN−1gN+OΔj,E64

and, for the adjoint matrix D†, we obtain

D†fj→Δj→0fj−fj−1Δj−γαfj+OΔj.E65

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

For a matrix with dimension 4×4, for instance, the eigenvalues are 0 of multiplicity two, 1Δ2−γα, and 1Δ3−γα, and the eigenvectors are

0,0,0,1T,E66

e−b1γα1−Δ2γα1−Δ3γαe−b2γα1−Δ3γαe−b3γα0T,E67

0−Δ3−Δ2e−b2γαΔ2e−b3γα0T,E68

0,0,1,0T,E69

and the complement, for arbitrary dimension, eigenvector is

0e−γb2/αα−ατ+γΔ2…e−γbN−1/α∏k=2N−2α−ατ+γΔk0T,E70

with eigenvalue τ∈C, and τ=1Δk−γα, k=2,3,…,N−1, the complement of the regular eigenvalues.

The determinant of the matrix 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, ajj=0. That matrix is

A2=0h1αh2Δ10…0000h2αh3Δ2…00000…00⋮000…0hN−1αhNΔN−1000…−hNαhN−1ΔN−10.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 αdiagΔj−1, with ΔN=ΔN−1.

The eigenvalues of the matrix A2 are

λ=0with multiplicityN−2,andλ±=±iαΔN−1,E72

and the corresponding eigenvectors are

xλ=100⋮0,ΔN−1N−2h1∏j=1N−2ΔjiΔN−1N−3h2∏j=2N−2Δj−ΔN−1N−4h3∏j=3N−2Δj⋮iN−2hN−1iN−1hN,−ΔN−1N−2h1∏j=1N−2Δj−iΔN−1N−3h2∏j=2N−2ΔjΔN−1N−4h3∏j=3N−2Δj⋮−iN−2hN−1−iN−1hN.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 β, and the eigenvector v with entries—for dimension five—given by

v5v4=−ih5h4,v4v3=h4βΔ3h3α,v3v2=h3βΔ2h2,v2v1=h2βΔ1h1α,v1∈C.E74

with eigenvalue β=±iα/Δ4.

The determinant of the matrix A2 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).

Jorge Toledano-Serrabona

Your cart