Available via license: CC BY

Content may be subject to copyright.

mathematics

Article

On the Inverse of the Caputo Matrix Exponential

Emilio Defez 1, Michael M. Tung 1,*, Benito M. Chen-Charpentier 2and José M. Alonso 3

1Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n,

46022 Valencia, Spain; edefez@imm.upv.es

2Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019-0408, USA;

bmchen@uta.edu

3Instituto de Instrumentación para Imagen Molecular, Universitat Politècnica de València, Camino de Vera,

s/n, 46022 Valencia, Spain; jmalonso@dsic.upv.es

*Correspondence: mtung@imm.upv.es or mtung@mat.upv.es

Received: 1 October 2019; Accepted: 18 November 2019; Published: 21 November 2019

Abstract:

Matrix exponentials are widely used to efﬁciently tackle systems of linear differential

equations. To be able to solve systems of fractional differential equations, the Caputo matrix

exponential of the index

α>

0 was introduced. It generalizes and adapts the conventional matrix

exponential to systems of fractional differential equations with constant coefﬁcients. This paper

analyzes the most signiﬁcant properties of the Caputo matrix exponential, in particular those related

to its inverse. Several numerical test examples are discussed throughout this exposition in order to

outline our approach. Moreover, we demonstrate that the inverse of a Caputo matrix exponential in

general is not another Caputo matrix exponential.

Keywords: Caputo matrix exponential; matrix inverse; fractional derivative

MSC: 15A09; 15A16

1. Introduction and Motivation

Formally a square matrix

A∈Cr×r

can be associated with its exponential matrix function

eAt

.

The traditional matrix exponential takes a prominent position among all matrix functions—ultimately

due to its relevance in the resolution of systems of ﬁrst-order ordinary differential equations. However,

in practice its efﬁcient numerical computation poses considerable difﬁculties, see [1] for details.

At the same time, systems of fractional differential equations, which contain derivatives extending

the standard integer-order derivative to arbitrary order

α≥

0, play an important role in many other

important applications of science and engineering [

2

–

4

]. Although fractional calculus is factually

known since the end of the 17th century [

5

], only during the recent decades its relevance for practical

modeling and engineering simulations has become evident. Fractional derivatives naturally implement

Volterra’s “principle of the dissipation of hereditary action”, meaning that causality aspects and

memory characteristics of dynamical systems may easily be incorporated. Important applications are

in hydrology, e.g., ﬂow simulations of ﬂuids in porous media, and in civil engineering, e.g., trafﬁc ﬂow

problems on road networks, among many others [6,7].

A great variety of fractional derivatives are proposed and used in the literature. The most

common fractional derivative is the derivative introduced by Caputo [

8

]. It is deﬁned in terms of the

Riemann–Liouville fractional integral of order α≥0 operating on function f(t):

Jαf(t) = 1

Γ(α)Zt

0(t−τ)α−1f(τ)dτ,t>0.

Mathematics 2019,7, 1137; doi:10.3390/math7121137 www.mdpi.com/journal/mathematics

Mathematics 2019,7, 1137 2 of 11

Then, provided that

f

is a locally integrable function, the following operation on

f

deﬁnes its

fractional derivative of Caputo with order α≥0:

Dαf(t) = Jm−αDmf(t),t>0, m−1<α≤m,m∈N.

Note that, as expected,

Dn=dn/dtn

agrees with the usual derivative of integer order

n∈N

.

(Our convention is to use Nfor the set of all positive integer numbers, whereas N0=N∪{0}.)

In 2016, Rodrigo [

9

] introduced the fractional exponential matrix of Caputo of order

α≥

0.

Similarly, here we are using the following deﬁnition for 0 ≤α≤1:

exp?(tαA;α)=∑

n≥0

Antαn

Γ(αn+1),t>0, (1)

in relation with the Mittag–Lefﬂer matrix function [10].

It is well-known that inverse problems [

11

] are among the most basic applications for the inverse

of the conventional exponential matrix. Moreover, mathematical optimisation is another area in which

the inverse of the matrix exponential is frequently encountered and of signiﬁcant relevance, see, e.g.,

(Reference [

12

], Equations (4.4) and (4.7)). Observe that if in a linear differential system all ordinary

derivatives are replaced by fractional derivatives of Caputo type, then the associated inverse problems

will necessarily involve the inverse of the Caputo matrix exponential.

In the case of the Caputo matrix exponential,

(1)

, there still remains to clarify the existence of its

inverse, in full analogy to the case of the conventional matrix exponential

eAt

with inverse matrix

e−At

.

The ﬁnal objective of this work will be to study the existence and computation of the inverse of the

Caputo matrix exponential.

The present paper is organized as follows. Section 2ﬁrst focuses on checking the main properties

of the matrix exponential of Caputo, and also on presenting counterexamples of other questionable

properties which eventually are not satisﬁed. In Section 3, we will demonstrate that, in general,

the inverse of an exponential matrix of Caputo is not another exponential matrix of Caputo. Finally,

Section 4concludes with the actual computation of the inverse of the Caputo matrix exponential and

gives examples.

In the remainder of this work, we will denote by

Cp×q

the set of rectangular complex matrices.

For a square matrix

A∈Cr×r

, as usual

σ(A)

denotes the spectrum of matrix

A

, i.e., the set of its

eigenvalues. Moreover, we will denote by

kAk

any multiplicative norm of matrix

A

. In particular,

kAk2is the 2-norm, deﬁned by

kAk2=sup

z6=0

kAzk2

kzk2

,

where for any vector

z∈Cq

, the usual Euclidean norm of

z

is

kzk2= (ztz)1/2

. Additionally, it will

be helpful to remember that for a family of matrices

A(k

,

n)∈Cr×r

with

n

and

k

being positive,

the following identity holds

∑

n≥0

∑

k≥0A(k,n) = ∑

n≥0

n

∑

k=0A(k,n−k). (2)

This identity is analogous to the one of the proof for Lemma 11 in (Reference [13], p. 57).

2. Caputo Matrix Exponential

This section ﬁrst presents some of the fundamental properties of the Caputo matrix exponential

which will be built upon in subsequent parts of this work. Then, the next subsection concentrates on

some striking counterexamples of properties which one could naively intuit but which at the end do

not hold. The ﬁnal subsection centers on the existence of the inverse of the Caputo matrix exponential

and its conditions.

Mathematics 2019,7, 1137 3 of 11

2.1. Properties

In the following, we list the most important fundamental properties of the Caputo matrix

exponential which unreservedly have to be fulﬁlled:

(a) For α=1, the Caputo matrix exponential coincides with the conventional matrix exponential:

exp?(tA; 1)=eAt . (3)

(b) If 0r×r,Ir×rare the null and identity matrices of Cr×r, respectively, it is clear that

exp?(0r×r;α)=Ir×r. (4)

(c)

If

A∈Cr×r

, and

σ(A)

denotes the set of its eigenvalues, it is well known that

A

has the Jordan

canonical factorization A=P J P−1, where Jis a diagonal block-matrix given by

J=diag {J1,J2, . . . , Jk},Ji=

λi10

λi

...

...1

0λi

,λi∈σ(A).

Then from Deﬁnition (1), it immediately follows that

exp?(tαA;α)=Pdiag exp?(tαJ1;α), exp?(tαJ2;α), . . . , exp?(tαJk;α)P−1. (5)

(d)

Avoiding entirely the Jordan canonical form of

A

and only knowing

σ(A)

, Putzer’s method

(see e.g., [9]) allows to explicitly obtain exp?(tαA;α)in fully analytical form.

2.2. Counterexamples

However, there are obvious differences between

exp?(tαA;α)

and the matrix exponential

eAt

,

which are straightforward to detect considering the scalar case r=1.

(1)

The matrix exponential

eAt

is a periodic function of period

T=

2

πi Ir×r

, where

i

as usual is the

imaginary unit:

eAt =eAt+2πi Ir×r.

However, this is not the case for the Caputo matrix exponential, even in the scalar case (

r=

1).

In fact, it easily can be checked that for A=1, t=1 and α=1/2, we have

exp?(1; 1/2)≈5.00898

exp?(1+2πi; 1/2)≈ −0.0144688 +0.0885799i,

so that

exp?(1; 1/2)6=exp?(1+2πi; 1/2).

Thus, we generally conclude that

exp?(A;α)6=exp?(A+2πiIr×r;α). (6)

(2) It is well known that if Aand Bare two commuting matrices, i.e., AB =BA, then

e(A+B)t=eAteBt. (7)

Mathematics 2019,7, 1137 4 of 11

This relation is generally not true for the fractional exponential matrix of Caputo—even for

the simplest scalar case (

r=

1). In fact, we can easily observe that when we take

A=B=1,

t=1, α=1/2, we have

exp?(1; 1/2)≈5.00898

exp?(2; 1/2)≈108.941,

so that

exp?(2; 1/2)6=exp?(1; 1/2)exp?(1; 1/2).

Consequently, we generally have

exp?(tα(A+B);α)6=exp?(tαA;α)exp?(tαB;α), (8)

and the Caputo matrix exponential therefore does not satisfy the semigroup property.

(3)

If we denote by

Det(A)

the determinant of the square matrix

A

and by

Tr(A)

its trace, i.e.,

the sum of the elements on the main diagonal, it is well known that the matrix exponential

satisﬁes

Det eA=eTr(A). (9)

In this way, it becomes obvious that the usual exponential matrix

eA

is always invertible, since

its determinant is always non-zero. Observe that the analogous identity for the Caputo matrix

exponential is not true, i.e.,

Det(exp?(A;α))6=exp?(Tr(A);α)

. To prove that this property is

not true, it is easy to check that

exp? tα 2−1

4−3!;α!=

4

3Eα(tα)−1

3Eα(−2tα)1

3Eα(−2tα)−1

3Eα(tα)

4

3Eα(tα)−4

3Eα(−2tα)4

3Eα(−2tα)−1

3Eα(tα)

,

where Eα(z)is the Mittag-Lefﬂer function deﬁned by

Eα(z) = ∑

j≥0

zj

Γ(αj+1). (10)

Now, taking t=1, α=1/2 and Tr 2−1

4−3!=−1, one gets that

Det exp? 2−1

4−3!; 1/2!!≈1.27927 , exp?(−1; 1/2)≈0.427584,

so that in general

Det(exp?(A;α))6=exp?(Tr(A);α).

(4) As a consequence of (7), it follows that for A∈Cr×rit is

eAte−At =Ir×r. (11)

For this reason the exponential matrix

eAt

is always invertible, and its inverse is precisely

e−At

.

On the other hand, for the inverse of the Caputo matrix exponential, it is easy to verify that

property (11) is not fulﬁlled.

As an example, we consider the two matrix exponentials

exp? tα 2−1

4−3!;α!=

4

3Eα(tα)−1

3Eα(−2tα)1

3Eα(−2tα)−1

3Eα(tα)

4

3Eα(tα)−4

3Eα(−2tα)4

3Eα(−2tα)−1

3Eα(tα)

Mathematics 2019,7, 1137 5 of 11

and

exp? −tα 2−1

4−3!;α!=

4

3Eα(−tα)−1

3Eα(2tα)1

3Eα(2tα)−1

3Eα(−tα)

4

3Eα(−tα)−4

3Eα(2tα)4

3Eα(2tα)−1

3Eα(−tα)

.

For the choice α=1/2 with t=1, we obtain

exp? 2−1

4−3!; 1/2!exp? − 2−1

4−3!; 1/2!≈ −6.41867 8.56043

−34.2417 36.3835 !6=I2×2,

and it clearly is

exp?(tαA;α)exp?(−tαA;α)6=Ir×r.

2.3. Existence

In order to guarantee the existence of the inverse of the Caputo matrix exponential

exp?(tαA;α)

for A∈Cr×r, observe that from deﬁnition (1), for α>0, t≥0, it follows that

Ir×r−exp?(tαA;α)

=

∑

n≥1

Antαn

Γ(αn+1)

≤∑

n≥1kAktαn

Γ(αn+1)=EαkAktα−1.

Then, according to Lemma 2.3.3 in (Ref. [

14

], p. 58), matrix

exp?(tαA;α)

is invertible in the interval

I= [0, t?], where

Eα(kAktα)<2. (12)

Taking into account that the Mittag–Lefﬂer function

g(t) = Eα(kAktα)

satisﬁes

g(

0

) =

1, and it

is a strictly increasing function for

t∈(

0,

+∞)

, we can conclude that there always exists

t?

so that

inequality (12) holds in I= [0, t?]. Therefore, exp?(tαA;α)−1also exists, at least for t∈I.

Example 1.

For the particular case

A= 2−1

4−3!

and also index

α=

1

/

4, it is easy to check that

kAk2≈5.46499. Then, it holds

E0.25 5.46499 t1/4<2⇔t∈[0, 0.0000594].

Thus, if t ∈[0, 0.0000594], the inverse exp? t1/4 2−1

4−3!; 1/4!−1

exists.

3. A New Inversion Property of the Caputo Matrix Exponential

In Section 2.3, we proved the existence of the inverse of the Caputo matrix exponential. It is

well-known that the inverse of the conventional matrix exponential

eAt

is again an exponential of

the matrix

−At

, or simply

eAt−1=e−At

. So can we arrive at a similar property for the Caputo

matrix exponential?

For this purpose, let us consider the matrix

A=

0100

0010

0001

0000

. (13)

Mathematics 2019,7, 1137 6 of 11

The square matrix

A

is a nilpotent matrix of index 3, meaning that

A36=

0

4×4

but

An=

0

4×4

for

n≥4. Thus, applying Deﬁnition (1), it is easy to establish

exp?(tαA;α)=

3

∑

n=0

Antαn

Γ(nα+1)=

1tα

Γ(α+1)t2α

Γ(2α+1)t3α

Γ(3α+1)

0 1 tα

Γ(α+1)t2α

Γ(2α+1)

0 0 1 tα

Γ(α+1)

0 0 0 1

.

Now, we proceed to calculate its inverse and obtain

exp?(tαA;α)−1=

1−tα

Γ(α+1)t2α

Γ(α+1)2−t2α

Γ(2α+1)−t3α

Γ(α+1)3+2t3α

Γ(α+1)Γ(2α+1)−t3α

Γ(3α+1)

0 1 −tα

Γ(α+1)t2α

Γ(α+1)2−t2α

Γ(2α+1)

0 0 1 −tα

Γ(α+1)

0 0 0 1

. (14)

Suppose that there exists a matrix B∈C4×4such that

exp?(tαA;α)−1=exp?(tαB;α)=∑

n≥0

Bntαn

Γ(nα+1). (15)

Then, we may recast the expression for the inverse given by (14) in the form

exp?(tαA;α)−1=I4×4−1

Γ(α+1)

0100

0010

0001

0000

tα+1

Γ(α+1)2−1

Γ(2α+1)

0010

0001

0000

0000

t2α

+−1

Γ(α+1)3+2

Γ(α+1)Γ(2α+1)−1

Γ(3α+1)

0001

0000

0000

0000

t3α

=I4×4−1

Γ(α+1)Atα+1

Γ(α+1)2−1

Γ(2α+1)A2t2α

+−1

Γ(α+1)3+2

Γ(α+1)Γ(2α+1)−1

Γ(3α+1)A3t3α.

(16)

Equating the powers of

tα

in (15) and (16), we observe that matrix

B

must satisfy the

following system

B=−A

B2=Γ(2α+1)1

Γ(α+1)2−1

Γ(2α+1)A2

B3=Γ(3α+1)−1

Γ(α+1)3+2

Γ(α+1)Γ(2α+1)−1

Γ(3α+1)A3

. (17)

Eliminating recursively all matrices of the previous system yields

Γ(2α+1)1

Γ(α+1)2−1

Γ(2α+1)=1

Γ(3α+1)−1

Γ(α+1)3+2

Γ(α+1)Γ(2α+1)−1

Γ(3α+1)=−1

. (18)

If the ﬁrst equation of (18) holds, i.e., α>0 satisﬁes

Γ(2α+1)

Γ(α+1)2=2, (19)

Mathematics 2019,7, 1137 7 of 11

then the second equation of (18) also holds. Equation (19) has the unique solution

α=

1, and therefore

also system (17). In consequence—except for the trivial case

α=

1—we afﬁrm that the inverse of the

Caputo matrix exponential generally is not another Caputo matrix exponential.

4. On the Computation of the Inverse of the Caputo Matrix Exponential

We now propose to determine the inverse of the Caputo matrix exponential. For this, we introduce

the following deﬁnition:

Deﬁnition 1.

Let

A∈Cr×r

be an arbitrary square matrix and

α>

0. We deﬁne the sequence of matrices

{Dn(α)}n≥0as

D0(α) = Ir×r,Dn(α) = −

n−1

∑

k=0

An−kDk(α)

Γ[(n−k)α+1],n≥1. (20)

We are now in the position to proceed with the following theorem, which is a reﬁnement of the

arguments already presented in Section 2.3, explaining why the inverse of

exp?(tαA;α)

exists for

t>

0,

though sufﬁciently small, and satisfying inequality (12).

Theorem 1.

Let

A∈Cr×r

be a square matrix and 0

<α≤

1. Let

t>

0be such that the fractional matrix

function f (t,α), deﬁned by

f(t,α) = ∑

n≥0

Dn(α)tnα, (21)

converges. Then, it holds

(a) exp?(tαA;α)f(t,α) = Ir×r,

(b) f (t,α)exp?(tαA;α)=Ir×r.

Proof of Theorem 1.For the proof of convergence of f(t,α), recall that asymptotically

Γ(nα+1)∼√2πe−αnαnαn+1

2,n→∞,

so that we conclude

lim

n→∞

Γ(nα+1)1/n

nα=e−ααα.

Hence, the series

∑

n≥1

znAn

Γ(nα+1)

converges for all

z∈C

. This convergence occurs uniformly on compact subsets of

C

. Therefore, the set

(z∈C: sup

λ∈σ(A)

∑

n≥1

znλn

Γ(nα+1)

<1)

contains a circular disc centered at the origin,

D

0,

r(A

,

α)

, with radius

r(A

,

α)>

0. This radius can be

determined by considering the Mittag-Lefﬂer function already introduced in

(10)

, speciﬁcally

Eαzλ

for z,λ∈C, which is analytic.

By the spectral mapping theorem, if

A∈Cr×r

, the spectrum

σEα(zA)

of

EαzA∈Cr×r

satisﬁes

σEα(zA)={Eα(zλ):λ∈σ(A)}.

Mathematics 2019,7, 1137 8 of 11

If Ais not a nilpotent matrix, then sup |λ|:λ∈σ(A)>0. In this case, we choose

rA,α=inf |z|:Eα(zλ) = 0 for some λ∈σ(A)

=inf |w|

|λ|:Eα(w) = 0, w∈C,λ∈σ(A)

=inf {|w|:Eα(w) = 0, w∈C}

sup |λ|:λ∈σ(A)>0

because

Eα(

0

) =

1. Moreover, the series

∑n≥0Dn(α)zn

converges for

|z|<rA

,

α

, because on the disc

D0, r(A,α)the matrix function EαzAis invertible and analytic in C.

If

A

is a nilpotent matrix, i.e.,

σ(A)={

0

}

, and has index

k∈N

, then

Dn(α)

deﬁned in

(20)

vanishes when n≥k. Thus, series (21) only has a ﬁnite number of terms and apparently converges.

Now, we proceed with the remainder of the Theorem 1, proving part (a).

(a) Applying the respective Deﬁnitions (1) and (21), we compute

exp?(tαA;α)f(t,α) = ∑

n≥0

Antnα

Γ(αn+1)! ∑

k≥0

Dk(α)tkα!

=∑

n≥0

∑

k≥0

AnDk(α)tkαtαn

Γ(αn+1), taking A(n,k) = AnDk(α)tkαtαn

Γ(αn+1)of (2),

=∑

n≥0

n

∑

k=0

An−kDk(α)tkαt(n−k)α

Γ[(n−k)α+1]=∑

n≥0 n

∑

k=0

An−kDk(α)

Γ[(n−k)α+1]!tnα

=D0(α) + ∑

n≥1 n

∑

k=0

An−kDk(α)

Γ[(n−k)α+1]!tnα

=Ir×r+∑

n≥1 n−1

∑

k=0

An−kDk(α)

Γ[(n−k)α+1]+Dn(α)!tnα

=Ir×r+∑

n≥1 n−1

∑

k=0

An−kDk(α)

Γ[(n−k)α+1]−

n−1

∑

k=0

An−kDk(α)

Γ[(n−k)α+1]!tnα

=Ir×r.

(b)

This equality is equivalent to (a), because the matrix operators

exp?(tαA;α)

and

f(t

,

α)

commute.

In fact, let

f(z)

and

g(z)

be holomorphic functions of the complex variable

z

, both deﬁned on an

open set

Ω⊂C

. Further, let matrix

A∈Cr×r

be such that

σ(A)∈Ω

. Then, from the properties

of the matrix functional calculus ([15], p. 558), it follows f(A)g(A) = g(A)f(A).

Remark 1.

Following the same line of argument of this proof, a result similar to that of Theorem 1is also valid

for bounded operators

A

acting in a complex Banach space. One requires the well-known fact that

σ(A)∈C

is

non-void and compact together with the equality

sup{|λ|:λ∈σ(A)}=lim sup

n→∞kAnk1/n.

Then, if A is quasi-nilpotent, it follows sup{|λ|:λ∈σ(A)}=0, and thus r(A,α) = ∞.

Mathematics 2019,7, 1137 9 of 11

Remark 2.

Note that the sequence

Dn(α)n∈N0

in Formula

(20)

may be recast into the following

compact expression

Dn(α) = Ann

∑

`=1

(−1)`∑

n1+...+n`=n,nj≥1

1

`

∏

j=1

Γ(αnj+1)

,

providing a closed form and thereby avoiding recurrence relations.

Obviously, for α=1, we have that (21) is the inverse of the usual matrix exponential:

Theorem 2. Let A ∈Cr×rbe a square matrix and α=1. Then, the matrix sequence deﬁned by (20) satisﬁes

Dn(1) = (−1)nAn

n!,n≥0. (22)

Proof of Theorem 2.

We proceed by mathematical induction on. For

α=

1, from deﬁnition (20),

one obtains for the base case (n=0):

D0(1) = Ir×r=(−1)0A0

0! .

In the same way, taking n=1, the deﬁnition of sequence Dn(1)immediately yields

D1(1) = −AD0(1)

1! =−A

1! =(−1)1A1

1! .

Finally, in the induction step, we suppose that for

k=

0, 1, 2,

. . .

,

n−

1, property (22) is true. Then,

for n, we conclude

Dn(1) = −

n−1

∑

k=0

An−kDk(1)

(n−k)!=−

n−1

∑

k=0

An−k(−1)kAk

(n−k)!k!=−An

n!

n−1

∑

k=0

(−1)kn!

(n−k)!k!=−An

n!

n−1

∑

k=0n

k(−1)k

=−An

n!"n

∑

k=0n

k(−1)k−n

n(−1)n#=−An

n!−n

n(−1)n=(−1)nAn

n!.

Now, we move on to the numerical computation of the inverse of the Caputo matrix exponential

evaluated in the previous example.

Example 2.

In Example 1, we have shown that the matrix inverse of the Caputo matrix exponential

exp?(tαA;α)

for

A= 2−1

4−3!

and

α=

1

/

4exists at least for

t

within the interval

I= [

0, 0.0000594

]

.

It is easy to verify that t =4×10−5∈I produces the following numerical result

exp? 4×10−51/4 2−1

4−3!; 1/4!= 1.177537946538906 −0.08207192162719279

0.32828768650877116 0.767178338402942 !,

and then

exp? 4×10−51/4 2−1

4−3!; 1/4!−1

= 0.8246349372113879 0.08821856737867267

−0.3528742695146907 1.2657277741047512 !.

Mathematics 2019,7, 1137 10 of 11

Evaluating only the ﬁrst 13 terms of the series (21) with f (0.00004, 0.25), we obtain

f13(4×10−5, 1/4) =

13

∑

k=0

Dk(1/4)(4×10−5)k/4 = 0.8246349372113879 0.08821856737867273

−0.3528742695146909 1.2657277741047515 !,

with an approximation error

exp? 4×10−51/4 2−1

4−3!; 1/4!−1

−f13(4×10−5, 1/4)

2

=3.1650 ×10−16.

Example 3.

Consider again the matrix

A

given in (13). In this case, the elements of the matrix sequence

{Dn(α)}n≥0can be calculated explicitly:

D0(α) = I4×4,

D1(α) = −A

Γ(α+1),

D2(α) = A21

Γ(α+1)2−1

Γ(2α+1),

D3(α) = A3−1

Γ(α+1)3+2

Γ(2α+1)Γ(α+1)−1

Γ(3α+1),

Dn(α) = 0for n ≥4.

Taking into account Deﬁnition (21), one simpliﬁes

f(t,α) = ∑

n≥0

Dn(α)tnα=

3

∑

n=0

Dn(α)tnα=D0(α) + D1(α)tα+D2(α)t2α+D3(α)t3α

=I4×4−A

Γ(α+1)tα+A21

Γ(α+1)2−1

Γ(2α+1)t2α

+A3−1

Γ(α+1)3+2

Γ(2α+1)Γ(α+1)−1

Γ(3α+1)t3α

=

1−tα

Γ(α+1)t2α

Γ(α+1)2−t2α

Γ(2α+1)−t3α

Γ(α+1)3+2t3α

Γ(α+1)Γ(2α+1)−t3α

Γ(3α+1)

0 1 −tα

Γ(α+1)t2α

Γ(α+1)2−t2α

Γ(2α+1)

0 0 1 −tα

Γ(α+1)

0 0 0 1

,

which gives the same result as the matrix inverse already calculated in (14).

5. Conclusions

The starting point of this discussion was the observation and well-known fact that the conventional

matrix exponential always possesses an inverse due to its semigroup property. On the other hand,

Caputo’s matrix exponential carries a leading role in fractional calculus.

In this work, we have shown that for the Caputo matrix exponential the inverse does not

necessarily exist per se. Nevertheless, its existence is guaranteed in a speciﬁc interval, which we

have determined to relate to an uncomplicated inequality, viz. Equation (12). Furthermore, we have

established that this inverse is generally not again a Caputo matrix exponential.

Additionally, several explicit procedures have been outlined to calculate the inverse of the Caputo

matrix exponential, and it is hoped that they will open up novel pathways for the development of

future numerical methods for its efﬁcient computation.

Mathematics 2019,7, 1137 11 of 11

Author Contributions:

E.D. contributed in funding acquisition, project administration, conceptualization,

investigation, formal analysis, methodology, validation, editing and writing; M.M.T. contributed in

conceptualization, investigation, methodology, validation, writing the original draft, writing and editing; B.M.C.-C.

and J.M.A. contributed in conceptualization, supervision, validation, visualization. All authors agree and approve

the ﬁnal version of this manuscript.

Funding:

This work has been partially supported by Spanish Ministerio de Economía y Competitividad and

European Regional Development Fund (ERDF) grants TIN2017-89314-P and by the Programa de Apoyo a la

Investigación y Desarrollo 2018 of the Universitat Politècnica de València (PAID-06-18) grant SP20180016.

Acknowledgments:

We wish to express our gratitude to the anonymous referees for their exceptionally useful

comments to improve the quality of the manuscript. Further, Emilio Defez acknowledges the kind reception by

the University of Texas at Arlington during a research stay in May 2019.

Conﬂicts of Interest:

The authors declare that there are no conﬂicts of interest regarding the publication of

this paper.

References

1.

Moler, C.; Van Loan, C. Nineteen dubious ways to compute the exponential of a matrix, twenty-ﬁve years

later. SIAM Rev. 2003,45, 3–49.

2.

Bologna, M. Short Introduction to Fractional Calculus; Charlas de Física de la Universidad de Tarapacá: Arica,

Chile, 2005; Volume 19, pp. 41–54.

3. Ortigueira, M.D.; Machado, J. What is a fractional derivative? J. Comput. Phys. 2015,293, 4–13.

4.

Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential

Equations, to Methods of Their Solution and Some of Their Applications; Academic Press: Cambridge, MD, USA,

1998; Volume 198.

5.

Oldham, K.; Spanier, J. The Fractional Calculus: Theory and Applications of Differentiation and Integration to

Arbitrary Order; Academic Press: Cambridge, MD, USA, 2016; Volume 111.

6.

Zhou, Y. Fractional Evolution Equations and Inclusions: Analysis and Control; Academic Press: San Diego, CA,

USA, 1974.

7.

Tarasov, V.E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media;

Springer Science & Business Media: Heidelberg, Germany, 2011.

8.

Caputo, M. Linear models of dissipation whose

Q

is almost frequency independent-II. Geophys. J. Int.

1967

,

13, 529–539.

9.

Rodrigo, M.R. On fractional matrix exponentials and their explicit calculation. J. Differ. Equ.

2016

,261, 4223–4243.

10.

Garrappa, R.; Popolizio, M. Computing the Matrix Mittag-Lefﬂer Function with Applications to Fractional

Calculus. J. Sci. Comput. 2018,77, 129–153.

11.

Denisov, A.M. Elements of the Theory of Inverse Problems; Inverse and Ill-Posed Problems Series; VSP: Utrecht,

The Netherlands, 1999.

12.

Koivunen, V.; Abrudan, T. Riemannian Optimization in Complex-Valued ICA. In Advances in Independent

Component Analysis and Learning Machines; Academic Press: San Diego, CA, USA, 2015.

13. Rainville, E.D. Special Functions; Macmillan: New York, NY, USA, 1960; Volume 442.

14.

Golub, G.H.; Van Loan, C.F. Matrix Computations; Johns Hopkins University Press: Baltimore, MD, USA, 1996.

15. Dunford, N.; Schwartz, J.T. Linear Operators. Part 1: General Theory; Interscience: New York, NY, USA, 1957.

c

2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access

article distributed under the terms and conditions of the Creative Commons Attribution

(CC BY) license (http://creativecommons.org/licenses/by/4.0/).