EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 2, 2022, 681-725
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Spectral dichotomy methods of a matrix with respect to
the general equation of the parabola

Seydou Traoré1,∗, Mouhamadou Dosso1

1 Laboratoire de de Mathématiques fondammatales et Applications, UFR Mathématiques et
Informatique, Université Félix Houphouët-Boigny, Abidjan, Côte d’Ivoire

Abstract. This paper presents methods of spectral dichotomy of a matrix which compute spectral
projectors on the subspace associated with the eigenvalues external to the parabolas described by
a general equation. These methods are modifications of the one proposed in [A. N. Malyshev and
M. Sadkane, SIAM J. MATRIX ANAL. APPL. 18 (2), 265-278, 1997] which uses the spectral
dichotomy Theoretical and method of a matrix with respect to the imaginary axis. algorithmic
aspects of the methods are developed. Numerical results obtained by applying methods presented
on matrices are reported.

2020 Mathematics Subject Classifications: 65F15, 34D09, 47A46

Key Words and Phrases: Spectral dichotomy method, spectral projector, eigensubspaces, eigen-
values.

1. Introduction

Let A ∈ Rn×n (n > 1) be a matrix and Γ(a, b, c) a parabola with an equation of the
type

x = ay2 + by + c a ̸= 0. (1)

The aim of this paper is to propose spectral dichotomy methods which partition the
spectrum of matrix A into two parts : A first part inside the parabola and a second one
outside. This will lead to the calculation of the projectors associated respectively with the
eigenvalues inside and outside the parabola.

Equation (1) reduces to the following form

x = a

[(
y +

b

2a

)2

− disc

4a2

]
(2)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4348

Email addresses: tsaid06@yahoo.fr (S. Traoré), mouhamadou.dosso@univ-fhb.edu.ci (M. Dosso)

https://www.ejpam.com 681 © 2022 EJPAM All rights reserved.



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 682

where disc = b2 − 4ac ; or again

1

a

(
x+

disc

4a

)
=

(
y +

b

2a

)2

(3)

Throughout this paper we assume that the coefficient a has a negative sign. Thus, the
parabola with equation (3) takes the form

2p(d− x) = (y − pb)2 (4)

by setting

p = − 1

2a
and d = −disc

4a
= −b2 − 4ac

4a

We assume that the matrix A has no eigenvalues on the parabola Γ(a, b, c) for any variation
of the parameters a, b and c with a ̸= 0. From the work done in [13, 15], we propose in this
paper spectral dichotomy methods which give the projector P on the subspace associated
with the eigenvalues located outside of Γ(a, b, c). The paper is organized as follows. Section
2 gives preliminaries used in the implementation of our proposed methods. It consists of
three subsections. The first subsection summarizes the methods of spectral dichotomy of
a matrix and a pencil of matrices with respect to a circle developed respectively by M.
Dosso and al. in [2, 4] and M. Sadkane and al. in [15]. The second subsection makes
a brief presentation of the spectral dichotomy method of a matrix with respect to the
imaginary axis (see [15]). The last subsection presents the study made by A.N.Malyshev
and M.Sadkane in [13]. Section 3 presents new methods of spectral dichotomy of a matrix
with respect to the curve Γ(a, b, c) for variations of parameters a, b and c with a ̸=
0. Finally in section 4, numerical tests are used on various examples to illustrate the
effectiveness of the methods presented.

Throughout this paper, the identity and zero matrices of order k are denoted by Ik
and 0k or just I and 0 whenever the order is clear from the context. The 2-norm of a
matrix A is denoted by ∥A∥.

2. Preliminaries on spectral dichotomy methods

2.1. Spectral dichotomy with respect to a circle

Let A be a matrix having no eigenvalues on the circle C(0, r) (where r > 0). The
spectral projector on the subspace corresponding to the eigenvalues inside the unit circle
is defined by

P =
1

2iπ

∫
C
(zIn −A)−1dz =

1

2π

∫ 2π

0

(
In − e−iθ

r
A

)−1

dθ (5)

The computation of the spectral projector is accompanied by that of the Hermitian matrix
defined by:



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 683

H = H(r) =
1

2π

∫ 2π

0

(
I − e−iθA

r

)−∗

H(0)

(
I − e−iθA

r

)−1

dθ, (6)

with H(0) = (H(0))∗ > 0, an arbitrary Hermitian positive definite matrix used for scaling
purpose.

Remark 1. The spectral norm of H indicates the behavior of the spectral projector P. The
smaller ∥H∥ is, better is the quality of the dichotomy.

The couple of matrices (P,H) is the only solution of the generalized Lyapunov’s equa-
tion [8] 

r2H−A∗HA = P∗H(0)P− (I − P)∗H(0)(I − P)
PA = AP
P2 = P
PH = (PH)∗

(7)

That generalized equation was first proposed by Godunov in partial form in [7] and later
by Bulgakov in complete form (7) in [1]. The most efficient numerical method for the
circular dichotomy was first proposed in [10] and [12].
Moreover, for any vector x and for any integer k, we have the estimates [4, 6, 14]

∥AkPx∥ ≤
√

∥H∥∥H−1∥∥
(
1− 1

∥H∥

) k
2 ∥x∥

∥AkPx∥ ≥ 1√
∥H∥∥H−1∥∥

(
1 + 1

∥H∥

) k
2 ∥(I − P)x∥

(8)

which shows the importance of the quantity ∥H∥ on asymptotic decay to 0 ( or growth to
+∞) of the powers of A.

Different authors have proposed methods for determining the projector P and the
matrix H. We summarize the most important steps of the method proposed in [2, 4].
Note that this method is a variant of an initial method proposed by S.K. Godunov and M.
Sadkane in [9]. During their work, these authors have given some important results. The

first proposition gives the link in the one hand between the sequences of matrices Z
(2j+1)
k

and Z
(2j)
k , and in the other hand between Hj+1 and Hj .

Proposition 1. For j = 0, 1, · · · and k = 0, 1, · · · , 2j, we have

Z
(2j+1)
k = Z

(2j)
k Kj+1 (9)

Z
(2j+1)
2j+k = Z

(2j)
k Lj+1 (10)

Hj+1 = (Kj+1)
∗HjKj+1 + (Lj+1)

∗HjLj+1. (11)

For the details of the proof, see in [2, 4].

In the second proposition, the sequences of matrices (Lk)k≥0 and (Kr)r≥0 are computed
iteratively



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 684

Proposition 2. For j = 0, 1, · · · , we have(
Bj Aj

Aj Bj

)
.

(
Kj+1

Lj+1

)
=

(
0
In

)
(12)

with Aj = −AZ
(2j)
1 , Bj = Z

(2j)

2j

For the details of the proof, see in [2, 4].

The third important result gives an estimate of the error Z
(2j+1)

2j+1 −P when j takes large
values

Theorem 1. There exists j0 ∈ N such that for all j ≥ j0, we have

∥Z(2j+1)

2j+1 − P∥ ≤ κ2(X)
ωγ2

j+1

1− ωγ2j+1 .

with ω > 1 and 0 < γ < 1.

For the details of the proof, see in [2, 4].

This last result shows the fast convergence of Z
(2j+1)

2j+1 to the projector P.
These results led to the following algorithm

Algorithm 1 (DichoC1).

• Input variables: A and In such that the matrix pencil zIn−A has no eigenvalues on
the circle C(O, r) with center O and the radius r.

• Output variables: The spectral projector P and the dichotomy criterion H.
P being the projector on the right invariant space of zIn − A corresponding to the
eigenvalues inside the circle C(O, r) and H the dichotomy criterion.

(i) Initialize

(a) A0 = −A

r
.

(b) resolve (
A0 In
In A0

)(
K1

L1

)
=

(
0
In

)
.

(c) Put Z
(2)
1 = K1, Z

(2)
2 = L1 and compute H1 = (Z

(2)
1 )∗(Z

(2)
1 ) + (Z

(2)
2 )∗(Z

(2)
2 ).

(ii) Iterate : For j = 1, 2, ...

(a) put

Aj = A0Z
(2j)
1 , Bj = Z

(2j)

2j
.



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 685

(b) Resolve (
BJ Aj

Aj Bj

)(
Kj+1

Lj+1

)
=

(
0
In

)
.

(c) Compute

Z
(2j+1)
1 =Z

(2j)
1 Kj+1

Z
(2j+1)

2j+1 =Z
(2j)

2j
Lj+1

Hj+1 =(Kj+1)
∗HjKj+1 + (Lj+1)

∗HjLj+1.

(iii)

P = Z
(2j+1)

2j+1 and H = Hj+1.

Another spectral dichotomy method has been proposed by M. Sadkane and A. Touhami
in [15]. We will just present the resulting algorithm of their work within the framework
of the spectral dichotomy method of a pencil λB −A

Algorithm 2 (DichoC2).

• Input: A,B ∈ Cn×n such that the pencil λB − A is regular having no eigenvalues
on the unit circle.
H(0) = (H(0))∗ used for scaling. For instance H(0) = In

• Output: P the spectral projector onto the right deflating subspace of λB − A asso-
ciated with the eigenvalues inside the unit circle.
H the matrix integral whose norm ∥H∥ indicates the quality of the projector P.

1. Initialization
H0 = H0

First iteration

(i) Compute X,Y solutions of the equations
X(B −A) = A, and Y (B −A) = B

(ii) Compute ∆0,∇0 solutions of the equations
(A+B)∆0 = X, and (A+B)∇0 = Y

(iii) Compute H1, Z
(2)
1 , Z

(2)
2 :

H1 = ∆∗
0H0∆+∇∗

0H0∇0

Z2
1 = ∆0, Z

(2)
2 = ∇0

2. Next Iterations
For j = 2, 3 · · · until convergence Do :
Update of Aj−1



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 686

(i) Aj−1 = −AZ
(2j−1)
1

(ii) Compute ∆j−1

(2Aj−1 − In)∆j−1 = Aj−1

Computation of Hj , Z
(2j)
1 , Z

(2j)

2j
:

(iii) Hj = ∆∗
j−1Hj−1∆j−1 + (I −∆j−1)

∗Hj−1(In −∆j−1)

(iv) Z
(2j)
1 = Z

(2j−1)
1 ∆j−1, Z

(2j)

2j
= Z

(2j−1)

2j−1 (In −∆j−1)

EndFOR

3. P = Z
(2j)

2j
B and H = Hj

2.2. Spectral dichotomy with respect to the imaginary axis

We assume that λIn−A does not have any eigenvalue on the imaginary axis. We sum-
marize the computation of the spectral projector on the right eigenspace corresponding to
the eigenvalues with positive real parts.

Using the Cayley transformation φ : λ ∈ C \ {1} −→ z ∈ C \ {1}, defined by

φ(λ) = z =
(λ+ 1)

(λ− 1)
(13)

φ is a bijection from C \ {1} to C \ {1}. The spectral dichotomy with respect to the
imaginary axis can be transformed to the spectral dichotomy to the circle by the inverse
of φ.
It is not difficult to show that the bijection φ transforms the interior (respectively exterior)
of the circle to the left (respectively the right) half plane and the circle to the imaginary
axis. We will briefly prove it below.
Let’s assume that z = x+ iy then

z =
ℜ(λ) + iℑ(λ) + 1

ℜ(λ) + iℑ(λ)− 1

=
ℜ(λ)2 − 1 + ℑ(λ)2 − 2iℑ(λ)

(ℜ(λ)− 1)2 + ℑ(λ)2



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 687

So

x =
ℜ(λ)2 − 1 + ℑ(λ)2

(ℜ(λ)− 1)2 + ℑ(λ)2
and y =

−2ℑ(λ)
(ℜ(λ)− 1)2 + ℑ(λ)2

we have:

λ ∈ C(O, 1) ⇔ |λ| = 1

⇔ ℜ(λ)2 + ℑ(λ)2 = 1

⇔ x = 0

which proves that φ maps bijectively the circle C(O, 1)\{(1, 0)} onto the imaginary axis.
Similarly we have,

|λ| < 1 ⇔ ℜ(λ)2 + ℑ(λ)2 < 1

⇔ x < 0

Consider the pencil λB − A where B = A− In and A = A+ In. The eigenvalues z of

the matrix A and λ are linked by the relation z =
λ+ 1

λ− 1
.

Therefore, the spectral dichotomy with respect to the imaginary axis can be transformed
to the spectral dichotomy to the unit circle and their spectral projectors are the same.
According to [3],[11],[12], the quality of the spectral dichotomy with respect to the imag-
inary axis is characterized by the numerical parameter

α = sup
ℜ(z)=0

∥(zIn −A)−1∥ (14)

Similarly, the quality of the dichotomy for the matrix pencil λB − A with respect to the
unit circle is also given by

α̃ = sup
|λ|=1

∥(λB −A)−1∥ (15)

The following proposition shows the relation between the two parameters.

Proposition 3. We assume that ∥A∥ = 1 and let α and α̃ be the two parameters defined
by (14) and (15). Then

1

2
α ≤ α̃ ≤ α+

1

2
(16)

Proof. Since

λB −A =
z + 1

z − 1
B − (A+ In)

=
1

z − 1
[(z + 1)(A− In)− (z − 1)(A+ In)]

=
−2

z − 1
(zIn −A)



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 688

then (λB −A)−1 = −z − 1

2
(zIn −A)−1 and thus

α̃ = sup
ℜ(z)=0

1

2
| z − 1 | ∥(zIn −A)−1∥ ≥ 1

2
α

We also have

∥ (λB −A)−1 ∥ ≤ 1

2
(1 + |z|)∥ (zIn −A)−1 ∥

We will discuss according to the values of |z|

• If |z| ≤ α+ 1

α

sup
|λ|=1

∥ (λB −A)−1 ∥ ≤ 1

2
sup

ℜ(z)=0
(1 + |z|)∥ (zIn −A)−1 ∥

≤ α+
1

2

then α̃ ≤ α+
1

2

• If |z| ≥ α+ 1

α
then with the assumption ∥A∥ = 1 it follows that

∥∥∥∥Az
∥∥∥∥ ≤ 1. Which

leads to

(zIn −A)−1 =
1

z

(
In − A

z

)−1

=
1

z

(
In +

A

z

+∞∑
m=0

Am

zm

)

=
1

z

(
In +A(zIn −A)−1)

)
Consequently, we obtain

∥(λB −A)−1∥ ≤ 1

2
(1 + |z|)∥(zIn −A)−1∥

≤ 1

2
(1 +

1

|z|
)
(
1 + ∥(zIn −A)−1∥

)
≤ 2α+ 1

2(α+ 1)

(
1 + ∥(zIn −A)−1∥

)
hence

α̃ ≤ 2α+ 1

2(α+ 1)
(1 + α) = α+

1

2



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 689

In both cases

α̃ ≤ α+
1

2

In conclusion
1

2
α ≤ α̃ ≤ α+

1

2

This proposition shows that the quality of the dichotomy of A with respect to the
imaginary axis is equivalent to the dichotomy of the matrix pencil λB − A with respect
to the unit circle. The following algorithm is used to calculate the values of the projector
and the dichotomy criterion

Algorithm 3 (DichoI).

• Input variables: A and In such that the matrix sheaf zIn −A has no eigenvalues on
the imaginary axis.

• Output variables: P and H.
P is the projector onto the left deflating subspace of A corresponding to the eigen-
values with real positives parts and H the dichotomy’s criterion.

1. Set A = A+ I and B = A− I.

2. Using Algorithm 2 to λB−A, compute the projectors Pi onto the right eigenspace of
A associated with the eigenvalues inside the unit circle and the Hermitian matrice
H.

3. P = In − Pi.

2.3. Spectral dichotomy with respect to a parabola

Consider the equation of the following parabola

2p
(p
2
− x
)
= y2 with p > 0 (17)

which was studied by Malyshev and Sadkane in [13]. We make a brief summary :
Consider the matrix A of order n (n > 1) having no eigenvalues on the parabola

Γ = Γ(a, 0, c) of Equation (17). Let A be the matrix of order 2n defined by

A =


−
√

p
2In A

In −
√

p
2In


The respective eigenvalues λ and z of the matrices A and A satisfy the relation



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 690

z =

(
λ+

√
p

2

)2

In their study, Malyshev and Sadkane assumed in ([13]) that ∥A∥ = 1. Otherwise we set

A1 =
1

∥A∥
A and p1 =

p

∥A∥
.

According to [13], the quality of the spectral dichotomy with respect to the imaginary
axis is characterized by the numerical value

αA = sup
ℜ(λ)=0

∥(λI2n −A)−1∥. (18)

Similarly, the spectral dichotomy with respect to the parabola is also characterized by the
numerical parameter

αA = sup
z∈Γ

∥(zIn −A)−1∥ (19)

The following proposition gives a relation between the parameters αA and αA.

Proposition 4. ([13]) Let αA and αA be the two parameters defined by (18) and (19).
We have

αA ≤ αA ≤ αA +
√
αA

√
1 + αA (20)

Consider the spectral projectors

• P ∈ Cn×n on the right eigenspace of A associated with the eigenvalues outside the
parabola Γ ;

• P ∈ C2n×2n on the right eigenspace of A associated with the eigenvalues in the right
complex half-plane.

The following proposition characterizes the relation between P and P

Proposition 5. ([13]) Consider a partition of the matrix P in the form

P =

(
P1 P2

P3 P4

)
with Pi ∈ Cn×n, i = 1, 4 (21)

Then
P = 2P1 = 2P4 = 4P2P3 (22)

Moreover

P2 =
1

2
(PA)

1
2 (23)



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 691

Algorithm 4 (DichoP).

• Input variables: A and In such that the matrix sheaf zIn −A has no eigenvalues on

the parabola of the equation 2p
(p
2
− x
)2

= y2 with p > 0

• Output variables: The spectral projector P and the dichotomy criterion H.
P being the projector on the right invariant space of zIn − A corresponding to the
eigenvalues outside the parabola and H the dichotomy criterion.

1. Compute the matrix

A =


−
√

p
2In A

In −
√

p
2In


2. Using Algorithm 3 to λI2n − A, compute the Projector P onto the right eigenspace

of A associated with the eigenvalues on the right half-plane of the complex plane and
the matrix H ;

3. If ∥H∥ is not large then determine the projector P by

P = 2P1

3. Presentation of new methods

In what follows, we will consider the general equation of the parabola (1) as announced
in the introduction and we will determine the projector P for parameters a, b and c ∈ R
with a ̸= 0.

3.1. The case of a parabola of equation of type (1) with discriminant
equal to 1.

Equation (1) of the parabola becomes

2p
(p
2
− x
)
= (y − pb)2 (24)

with p > 0.
For the parameter b = 0, we are in the case of the parabola studied by Malyshev and
Sadkane in [13].
On the other hand, in this section, we are going to consider the coefficient b ̸= 0. Which
leads us to define the following parabola

Γ̃ = {z = x+ iy \ x+ i(y − pb) ∈ Γ}



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 692

of the equation

2p
(p
2
− x
)
= ỹ2 (25)

where

ỹ = y − pb.

Consider the matrix of order 2n defined by

à =


−
√

p

2
In Ab

In −
√

p

2
In


where Ab = A− ipbIn

Remark 2. The respective eigenvalues λ̃ and z of the matrices à and A are such that

z =

(
λ̃+

√
p

2

)2

+ ipb

We also assume that ∥Ab∥ = 1. Otherwise (i.e. ∥Ab∥ ≠ 1), we can take

A1
b =

1

∥Ab∥
Ab and p1 =

1

∥Ab∥
p.

Consider the dichotomy quantities characterized by the following numerical parameters

αà = sup
ℜ(λ̃)=0

∥(λ̃I2n − Ã)−1∥ and αAb
= sup

z∈Γ̃
∥(zIn −A)−1∥ (26)

We have the following proposition

Proposition 6. Let αà and αAb
be the two parameters defined in (26). Assume that

∥Ab∥ = 1 and |pb| < 1

αAb

. (27)

Then

αAb
≤ αà ≤ 2

(
αAb

+
√
αAb

(
1 +

√
αAb

+ 1
))

(28)

Proof. Consider the matrix

(λ̃I2n − Ã) =


(λ̃+

√
p
2)In −Ab

−In (λ̃+
√

p
2)In

 .



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 693

We have


(λ̃+

√
p
2)In Ab

In (λ̃+
√

p
2)In

×


(λ̃+

√
p
2)In −Ab

−In (λ̃+
√

p
2)In

 =


(λ̃+

√
p
2)

2In −Ab 0

0 (λ̃+
√

p
2)

2In −Ab


Therefore

(λ̃I2n − Ã)−1 =


(λ̃+

√
p
2)

2In −Ab 0

0 (λ̃+
√

p
2)

2In −Ab


−1

×


(λ̃+

√
p
2)In Ab

In (λ̃+
√

p
2)In



=


((λ̃+

√
p
2)

2 + ipb)In −A 0

0 ((λ̃+
√

p
2)

2 + ipb)In −A


−1

×


(λ̃+

√
p
2)In A− ipbIn

In (λ̃+
√

p
2)In



=

(zIn −A)−1 0

0 (zIn −A)−1

×

√z − ipbIn A− ipbIn

In
√
z − ipbIn



=

√z − ipb(zIn −A)−1 (zIn −A)−1(A− ipbIn)

(zIn −A)−1
√
z − ipb(zIn −A)−1


Knowing that the norm (λ̃I2n − Ã)−1 is greater than or equal to the norm of each of

its block components taken individually, we can deduce that

α̃ = sup
ℜ(λ̃)=0

∥(λ̃I2n − Ã)−1∥ ≥ sup
z∈Γ̃

∥(zIn −A)−1∥ = αAb



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 694

and also

∥(λ̃I2n − Ã)−1∥ ≤

∥∥∥∥∥∥
√

z − ipbIn A− ipbIn

In
√
z − ipbIn

∥∥∥∥∥∥ ∥(zIn −A)−1∥

≤

∥∥∥∥∥∥
∥

√
z − ipbIn∥ ∥A− ipbIn∥

∥In∥ ∥
√
z − ipbIn∥

∥∥∥∥∥∥ ∥(zIn −A)−1∥

≤

∥∥∥∥∥∥

√

|z|+
√

|pb| 1

1
√

|z|+
√
|pb|

∥∥∥∥∥∥ ∥(zIn −A)−1∥

≤
(
(1 +

√
|z|+

√
|pb|
)
∥(zIn −A)−1∥

• If |z| ≤ αAb
+ 1

αAb

then

∥(λI2n − Ã)−1∥ ≤ αAb
(
√
|z|+

√
|pb|+ 1)

≤ αAb

(√
αAb

+ 1

αAb

+

√
1

αAb

+ 1

)
≤ αAb

+
√
αAb

(
1 +

√
αAb

+ 1
)

• If |z| > αAb
+ 1

αAb

then with the assumptions ∥Ab∥ = 1 and |pb| < 1

αAb

we note that

∥∥∥∥Az
∥∥∥∥ <

αAb

αAb
+ 1

(∥Ab∥+ |pb|)

<
αAb

αAb
+ 1

(
1 +

1

αAb

)
< 1

Which leads to

(zIn −A)−1 =
1

z

(
In − A

z

)−1

=
1

z

(
In +

A

z

+∞∑
m=0

Am

zm

)

=
1

z

(
In +A(zIn −A)−1)

)
Therefore



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 695

∥(λ̃I2n − Ã)−1∥ ≤
∥∥∥∥1z In +

1

z
A (zIn −A)−1

∥∥∥∥× ∥∥∥1 +√|z|+
√
|pb|
∥∥∥

≤ (∥A∥∥(zIn −A)−1∥+ 1)
1 +

√
|z|+

√
|pb|

|z|

≤ ((1 + |pb|)αAb
+ 1)

(
αAb

αAb
+ 1

+

√
αAb√

αAb
+ 1

+
αAb

√
|pb|

αAb
+ 1

)

≤ ((1 +
1

αAb

)αAb
+ 1)

(
αAb

αAb
+ 1

+

√
αAb√

αAb
+ 1

+

√
αAb

αAb
+ 1

)

≤ 2 + αAb

αAb
+ 1

(
αAb

+
√
αAb

√
αAb

+ 1 +
√
αAb

)

≤ 2
(
αAb

+
√
αAb

(
1 +

√
αAb

+ 1
))

.

Hence
αAb

≤ α̃ ≤ 2
(
αAb

+
√
αAb

(
1 +

√
αAb

+ 1
))

.

This proves that the dichotomy parameters of a matrix with respect to a parabola and
with respect to the imaginary axis are equivalent.

Consider spectral projectors

• P̃ ∈ Cn×n on the right subspace of A associated with its eigenvalues outside the
parabola Γ̃.

• P̃ ∈ C2n×2n on the right subspace of à associated with its eigenvalues in the complex
right half-plane.

We obtain the following proposition which characterizes the relation between P̃ and P̃

Proposition 7. Consider a partition of the matrix P̃ in the form

P̃ =

(
P̃1 P̃2

P̃3 P̃4

)
avec P̃i ∈ Cn×n, i = 1, 4 (29)

Then

P̃ = 2P̃1 = 2P̃4 = 4P̃2P̃3 (30)



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 696

Moreover

P̃A = 4× (P̃2)
2 + 2ipbP̃1 (31)

Proof.
Let X̃ be a solution to the matrix equation(

X̃ +

√
p

2
In

)2

= Ab (32)

Consider the matrix X̃1 defined by

X̃1 = −X̃ − 2

√
p

2
In.

We notice that

(
X̃1 +

√
p

2
In

)2

=

(
X̃ +

√
p

2
In

)2

= Ab

Hence X̃1 is also a solution of matrix equation (32).

X̃ +
√

p
2In −X̃ −

√
p
2In

In In

×

X̃ 0

0 −X̃ − 2
√

p
2In

×


1
2(X̃ +

√
p
2In)

−1 1
2In

−1
2(X̃ +

√
p
2In)

−1 1
2In



=


X̃(X̃ +

√
p
2In) (−X̃ −

√
p
2In)(−X̃ − 2

√
p
2In)

X̃ −X̃ − 2
√

p
2In

×


1
2(X̃ +

√
p
2In)

−1 1
2In

−1
2(X̃ +

√
p
2In)

−1 1
2In



=


1
2X̃ − 1

2(X̃ + 2
√

p
2In)

1
2(X̃ +

√
p
2In)(2X̃ + 2

√
p
2In)

1
2(X̃ +

√
p
2In)

−1(2X̃ + 2
√

p
2In)

1
2X − 1

2(X + 2
√

p
2In)



=


−
√

p
2In Ab

In −
√

p
2In





S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 697

= Ã.

Let X̃ = Q

[
J+ 0
0 J−

]
Q−1 be the canonical jordan form of the matrix X̃ with J+ and

J− the Jordan blocks associated respectively with the eigenvalues located in the right
half-plane and the left half-plane where J+ if of order k.

By replacing the decomposition of X̃ in the matrix Ã, we get

à =

Q
[
J+ 0
0 J−

]
Q−1 +

√
p

2
In −Q

[
J+ 0
0 J−

]
Q−1 −

√
p

2
In

In In

×


Q

[
J+ 0
0 J−

]
Q−1 0

0 Q

[
J+ 0
0 J−

]
Q−1 − 2

√
p

2
In

×


1

2

(
Q

[
J+ 0
0 J−

]
Q−1 +

√
p

2
In

)−1
1

2
In

−1

2

(
Q

[
J+ 0
0 J−

]
Q−1 −

√
p

2
In

)−1
1

2
In



=

Q 0

0 Q



[
J+ 0
0 J−

]
+

√
p

2
In −

[
J+ 0
0 J−

]
−
√

p

2
In

In In

×


[
J+ 0
0 J−

]
0

0

[
J+ 0
0 J−

]
− 2

√
p

2
In

×


1

2

([
J+ 0
0 J−

]
+

√
p

2
In

)−1
1

2
In

−1

2

([
J+ 0
0 J−

]
−
√

p

2
In

)−1
1

2
In


Q−1 0

0 Q−1



=Q̃J Q̃−1

with



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 698

Q̃ =

Q 0

0 Q



(
J+ 0
0 J−

)
+

√
p

2
In −

[
J+ 0
0 J−

]
−
√

p

2
In

In In


and

J =


[
J+ 0
0 J−

]
0

0

[
J+ 0
0 J−

]
− 2

√
p

2
In


Knowing that we have k eigenvalues of the matrix Ab in the right half-plane, we can

therefore compute the associated projector

P̃ = Q̃
[
Ik 0
0 0

]
Q̃−1

=

Q 0

0 Q



[
J+ 0
0 J−

]
+

√
p

2
In −

[
J+ 0
0 J−

]
−
√

p

2
In

In In

×


[
Ik 0
0 0

] [
0 0
0 0

]
[
0 0
0 0

] [
0 0
0 0

]


×



1

2

([
J+ 0
0 J−

]
+

√
p

2
In

)−1
1

2
In

−1

2

J+ 0

0 J−

+

√
p

2
In

−1

1

2
In


Q−1 0

0 Q−1



=

Q 0

0 Q




J+ +

√
p

2
Ik 0

0 0

 [
0 0
0 0

]
[
Ik 0
0 0

] [
0 0
0 0

]



×



1

2

(J+ +

√
p

2
Ik)

−1 0

0 (J− +

√
p

2
In−k)

−1

 1

2

[
Ik 0
0 In−k

]

−1

2

(J+ +

√
p

2
Ik)

−1 0

0 (J− +

√
p

2
In−k)

−1

 1

2

[
Ik 0
0 In−k

]



Q−1 0

0 Q−1





S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 699

=

Q 0

0 Q




1

2

[
Ik 0
0 0

]
1

2

J+ +

√
p

2
Ik 0

0 0


1

2

(J+ +

√
p

2
Ik)

−1 0

0 0

 1

2

[
Ik 0
0 0

]

Q−1 0

0 Q−1



=


Q

[1
2
Ik 0

0 0

]
Q−1 Q

12(J+ +

√
p

2
Ik) 0

0 0

Q−1

Q

12(J+ +

√
p

2
Ik)

−1 0

0 0

Q−1 Q

[1
2
Ik 0

0 0

]
Q−1



=

[
P̃1 P̃2

P̃3 P̃4

]

It follows that

P̃1 = Q

[1
2
Ik 0

0 0

]
Q−1 =

1

2
P̃

P̃2 = Q
1

2

J+ +

√
p

2
Ik 0

0 0

Q−1

P̃3 = Q
1

2

(J+ +

√
p

2
Ik)

−1 0

0 0

Q−1


=⇒ Pb = 4P̃2P̃3

P̃4 = Q

[1
2
Ik 0

0 0

]
Q−1 =

1

2
Pb

We also note that with

A = Ab + ipbIn

= Q


(
J+ +

√
p

2
Ik

)2

+ ipbIk 0

0

(
J− +

√
p

2
In−k

)2

+ ipbIn−k

Q−1



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 700

we have

P̃A = Q

[
Ik 0
0 0

]
Q−1 ×Q


(
J+ +

√
p

2
Ik

)2

+ ipbIk 0

0

(
J− +

√
p

2
In−k

)2

+ ipbIn−k

Q−1

= Q

(J+ + (

√
p

2
Ik

)2

+ ipbIk 0

0 0

Q−1

= 4P̃2
2 + 2ipbP̃1

Thus we obtain the equality (30) and (31)

Remark 3. if the parameter b = 0, whence equalities (31) are reduced to those of (23)

Algorithm 5 (DichoPb).

• Input variables : A and In such that the matrix bundle zIn − A has no eigenvalues

on the parabola with equation 2p
(p
2
− x
)2

= (y − pb)2 with p > 0

• Output variables: P̃ and H̃
P̃ being the projector on the right subspace of zIn−A associated with the eigenvalues
outside the parabola and the matrix H̃ whose norm gives the dichotomy criterion.

1. Determine the matrix

à =


−
√

p
2In A− ipbIn

In −
√

p
2In


2. Using Algorithm 3 to λ̃I2n − Ã, compute the Projector P̃ onto the right eigenspace

of à associted with the eigenvalues on the right half-plane of the complex plane and
the matrix H̃ ;

3. If ∥H̃∥ is not large, determine the projectors P̃ by

P̃ = 2P̃1



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 701

3.2. The case of a parabola of equation of type (1) with discriminant
different from 1

We consider the following change of variable in equation (4)

x̃ = x+
p

2
− d

We get

2p
(p
2
− x̃
)
= (y − pb)2 (33)

3.2.1. The spectral dichotomy method with the coefficient b = 0

Consider the set

Γd =
{
zd = x+ iy/x+ (

p

2
− d) + iy ∈ Γ

}
described by the following equation

y2 = 2p(
p

2
− x̃).

Let the matrix

Ad =

−
√

p

2
In Ad

In −
√

p

2
In

 where Ad = A+
(p
2
− d
)
In (34)

Knowing that the eigenvalues zd and z of the matrices Ad and A are linked by

zd = z +
p

2
− d,

Remark 4. The respective eigenvalues λd and z of the matrices Ad and A are such that

z =

(
λd +

√
p

2

)2

− p

2
+ d

Furthermore, since z = x+ iy, then we have
x =

(
ℜ(λd) +

√
p

2

)2

−ℑ(λd)
2 − p

2
+ d

y = 2

(
ℜ(λd) +

√
p

2

)
ℑ(λd)

By setting that



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 702

pd = 2

(
ℜ(λd) +

√
p

2

)2

then

y2 = 2pd

[pd
2

− x− p

2
+ d
]
= 2pd

(pd
2

− x̃
)

We also assume that ∥Ad∥ = 1. Otherwise we set (i.e. ∥Ad| ≠ 1), we can take

A1
d =

1

∥Ad∥
Ad and p1 =

1

∥Ad∥
.

Consider the numerical parameters αAd
and αAd

defined by

αAd
= sup

ℜ(λd)=0
∥(λdI2n −Ad)

−1∥ et αAd
= sup

z∈Γd

∥(zIn −A)−1∥ (35)

The following proposition gives a relation between the parameters αAd
and αAd

.

Proposition 8. Let αAd
and αAd

be the two parameters defined in (35). Assume that

∥Ad∥ = 1 and
∣∣∣p
2
− d
∣∣∣ < 1

αAd

. (36)

Then
αAd

≤ αAd
≤ 2

(
αAd

+
√
αAd

(
1 +

√
αAd

+ 1
))

. (37)

Proof.
Let the matrix

(λdI2n −Ad) =


(λd +

√
p
2)In −Ad

−In (λd +
√

p
2)In


We have


(λd +

√
p
2)In Ad

In (λd +
√

p
2)In

×


(λd +

√
p
2)In −Ad

−In (λd +
√

p
2)In

 =


(λd +

√
p
2)

2In −Ad 0

0 (λd +
√

p
2)

2In −Ad


with



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 703

(λdI2n −Ad)
−1 =


(λd +

√
p
2)

2In −Ad 0

0 (λd +
√

p
2)

2In −Ad


−1

×


(λd +

√
p
2)In Ad

In (λd +
√

p
2)In



=


(λd +

√
p
2)

2In − (A+ (p2 − d)In) 0

0 (λd +
√

p
2)

2In − (A+ (p2 − d)In)


−1

×


(λd +

√
p
2)In (A+ (p2 − d)In)

In (λd +
√

p
2)In



=


(λd +

√
p
2)

2 − p
2 + d)In −A 0

0 ((λd +
√

p
2)

2 − p
2 + d)In −A


−1

×


(λd +

√
p
2)In A+ (p2 − d)In

In (λd +
√

p
2)In



=

(zIn −A)−1 0

0 (zIn −A)−1

×


√
z + p

2 − dIn A+ (p2 − d)In

In

√
z + p

2 − dIn



=


√
z + p

2 − d(zIn −A)−1 (zIn −A)−1(A+ (p2 − d)In)

(zIn −A)−1
√

z + p
2 − d(zIn −A)−1


Knowing that the norm of (λdI2n−Ad)

−1 is greater than or equal to the norm of each
of its block components taken individually, we can deduce that



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 704

αAd
= sup

ℜ(λd)=0
∥(λdI2n −Ad)

−1∥ ≥ sup
z∈Γd

∥(zIn −A)−1∥ = αAd

and also

∥∥(λdI2n −Ad)
−1
∥∥ ≤

∥∥∥∥∥∥∥∥

√
z + p

2 − dIn A+ (p2 − d)In

In

√
z + p

2 − dIn


∥∥∥∥∥∥∥∥
∥∥(zIn −A)−1

∥∥

≤

∥∥∥∥∥∥∥∥

∥
√
z + p

2 − dIn∥ ∥A+ (p2 − d)In∥

∥In∥ ∥
√
z + p

2 − dIn∥


∥∥∥∥∥∥∥∥
∥∥(zIn −A)−1

∥∥

≤

∥∥∥∥∥∥∥∥

√
|z|+

√
|p2 − d| 1

1
√
|z|+

√
|p2 − d|


∥∥∥∥∥∥∥∥
∥∥(zIn −A)−1

∥∥

≤ (
√

|z|+
√

|p
2
− d|+ 1)

∥∥(zIn −A)−1
∥∥

• If |z| ≤ αAd
+ 1

αAd

then

∥(λI2n −Ad)
−1∥ ≤ αAd

(√
αAd

+ 1

αAd

+

√∣∣∣p
2
− d
∣∣∣+ 1

)

≤ αAd

(
1 +

√
1

αAd

)
+

√
αAd

√
1 + αAd

≤ αAd
+
√
αAd

(
1 +

√
αAd

+ 1
)

• If |z| > αAd
+ 1

αAd

with the conditions (36) we have

∥∥∥∥Az
∥∥∥∥ < 1.

Which leads to

(zIn −A)−1 =
1

z
(In − A

z
)−1

=
1

z

+∞∑
k=0

Ak

zk



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 705

=
1

z

(
In +

A

z

+∞∑
m=0

Am

zm

)

=
1

z

(
In +

A

z
(In − A

z
)−1

)
Consequently

∥(λdI2n −Ad)
−1∥ ≤

∥∥∥∥1z In +
1

z
A (zIn −A)−1

∥∥∥∥× (1 +√|z|+
√
|p
2
− d|

)

≤ ∥A(zIn −A)−1 + In∥ ×

 1

|z|
+

1√
|z|

+

√
|p
2
− d|

|z|



≤
((

1 +
∣∣∣p
2
− d
∣∣∣)αAd

+ 1
)
×

(
αAd

αAd
+ 1

+

√
αAd√

αAd
+ 1

+

√
αAd

αAd
+ 1

)

≤ (2 + αAd
)

αAd
+ 1

(
αAd

+
√
αAd

√
αAd

+ 1 +
√
αAd

)

≤ 2
(
αAd

+
√
αAd

(
1 +

√
αAd

+ 1
))

.

Consider spectral projectors

• Pd ∈ Cn×n on the right eigensubspace associated with the eigenvalues of A outside
the parabola Γd

• Pd ∈ C2n×2n on the right eigensubspace associated to the eigenvalues of Ad in the
right complex half-plane.

The following proposition characterizes the relation between Pd and Pd

Proposition 9. Consider a partition of the matrix Pd in the form

Pd =

(
P(d)
1 P(d)

2

P(d)
3 P(d)

4

)
with P(d)

i ∈ Cn×n, i = 1, 4 (38)

Then



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 706

Pd = 2P(d)
1 = 2P(d)

4 = 4P(d)
2 P(d)

3 (39)

Moreover
PdA = 4(P(d)

2 )2 − (p− 2d)P(d)
1 (40)

Proof.
Let Xd be a solution of the matrix equation(

Xd +

√
p

2
In

)2

= Ad. (41)

Following the same calculation as in the proof of Proposition 7, we get

Ad =

Xd +
√

p
2In −Xd −

√
p
2In

In In

×

Xd 0

0 −Xd − 2

√
p

2
In

×


1

2
(Xd +

√
p

2
In)

−1 1

2
In

−1

2
(Xd +

√
p

2
In)

−1 1

2
In


Let Xd = Qd

[
M+ 0
0 M−

]
Q−1

d be the canonical Jordan form of the matrix Xd with

M+ and M− the Jordan blocks associated respectively with the eigenvalues of Xd located
in the right half-plane and the left half-plane.

By replacing the decomposition of Xd in the matrix Ad, we get

Ad = Q̃dM(Q̃d)
−1

with

Q̃d =

Qd 0

0 Qd



(
M+ 0
0 M−

)
+

√
p

2
In −

[
M+ 0
0 M−

]
−
√

p

2
In

In In


and

M =


[
M+ 0
0 M−

]
0

0

[
M+ 0
0 M−

]
− 2

√
p

2
In


Therefore we can compute the associated projector

Pd = (Q̃d)

[
Ik 0
0 0

]
(Q̃d)

−1



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 707

=

Qd 0

0 Qd



[
M+ 0
0 M−

]
+

√
p

2
In −

[
M+ 0
0 M−

]
−
√

p

2
In

In In

×


[
Ik 0
0 0

] [
0 0
0 0

]
[
0 0
0 0

] [
0 0
0 0

]


×



1

2

([
M+ 0
0 M−

]
+

√
p

2
In

)−1
1

2
In

−1

2

M+ 0

0 M−

−
√

p

2
In

−1

1

2
In


Q−1

d 0

0 Q−1
d



=

Qd 0

0 Qd




M+ +

√
p

2
Ik 0

0 0

 [
0 0
0 0

]
[
Ik 0
0 0

] [
0 0
0 0

]



×



1

2

(M+ +

√
p

2
Ik)

−1 0

0 (M− +

√
p

2
In−k)

−1

 1

2

[
Ik 0
0 In−k

]

−1

2

(M+ +

√
p

2
Ik)

−1 0

0 (M− +

√
p

2
In−k)

−1

 1

2

[
Ik 0
0 In−k

]



Q−1
d 0

0 Q−1
d



=

Qd 0

0 Qd




1

2

[
Ik 0
0 0

]
1

2

M+ +

√
p

2
Ik 0

0 0


1

2

(M+ +

√
p

2
Ik)

−1 0

0 0

 1

2

[
Ik 0
0 0

]

Q−1

d 0

0 Q−1
d



=


Qd

[1
2
Ik 0

0 0

]
Q−1

d Q

12(M+ +

√
p

2
Ik) 0

0 0

Q−1
d

Qd

12(M+ +

√
p

2
Ik)

−1 0

0 0

Q−1
d Qd

[1
2
Ik 0

0 0

]
Q−1





S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 708

=

[
P(d)
1 P(d)

2

P(d)
3 P(d)

4

]
It follows that

P(d)
1 = Qd

[1
2
Ik 0

0 0

]
Q−1

d =
1

2
Pd

P(d)
2 = Qd

1

2

M+ +

√
p

2
Ik 0

0 0

Q−1
d

P(d)
3 = Qd

1

2

(M+ +

√
p

2
Ik)

−1 0

0 0

Q−1
d


=⇒ Pd = 4P(d)

2 P(d)
3

P(d)
4 = Qd

[1
2
Ik 0

0 0

]
Q−1

d =
1

2
Pd

With Xd = Q

[
M+ 0
0 M−

]
Q−1

d we have

A = Ad −
(p
2
− d
)
In

= Qd


(
M+ +

√
p

2
Ik

)2

−
(p
2
− d
)
Ik 0

0

(
M− +

√
p

2
In−k

)2

−
(p
2
− d
)
In−k

Q−1

and

PdA = Q

[
Ik 0
0 0

]
Q−1

d ×Q


(
M+ +

√
p

2
Ik

)2

− (
p

2
− d)Ik 0

0

(
M− +

√
p

2
In−k

)2

− (
p

2
− d)In−k

Q−1
d

= Qd

(M+ +

√
p

2
Ik

)2

− (
p

2
− d)Ik 0

0 0

Q−1
d

= 4(P(d)
2 )2 − (p− 2d)P(d)

1

Hence (39) and (40).

Remark 5. If the parameter d =
p

2
, whence equalities (40) are reduced to those of equal-

ities (23).



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 709

Algorithm 6 (DichoPd).

• Input variables: the matrices A , In and the real numbers d and p such that the matrix
pencil zIn−A has no eigenvalues on the parabola with equation 2p (d− x) = y2 with
p > 0 et d > 0

• Output variables: Pd and Hd.
Pd being the projector on the right subspace of zIn−A associated with the eigenvalues
outside the parabola and Hd the matrix whose norm defines the dichotomy criterion.

1. Determine the matrix

Ad =


−
√

p

2
In A+ (

p

2
− d)In

In −
√

p

2
In


2. Using Algorithm 3 to λdI2n−Ad, compute the Projector Pd onto the right eigenspace

of Ad associted with the eigenvalues on the right half-plane of the complex plane and
the matrix Hd.

3. If ∥Hd∥ is not large, determine the projector Pd by

Pd = 2P(d)
1 .

3.2.2. The spectral dichotomy method with the coefficient b ̸= 0

Consider the set

Γ̃d =
{
z = x+ iy/x+

(p
2
− d
)
+ i(y − pb) ∈ Γ

}
described by the following equation (33).
We consider the following order matrix 2n

Ãd =

−
√

p

2
In Adb

In −
√

p

2
In

 with Adb = A+
(p
2
− d− ipb

)
In.

The respective eigenvalues λ̃d and zdb of the matrices Ãd and Adb verify the relationship

zdb =

(√
p

2
+ λ̃d

)2

.

This leads to the following remark



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 710

Remark 6. The respective eigenvalues λ̃d and z of the matrices Ãd and A satisfy the
relation

z =

(√
p

2
+ λ̃d

)2

−
(p
2
− d− ipb

)
Furthermore, we get

x =

(
ℜ(λ̃d) +

√
p

2

)2

−ℑ(λ̃d)
2 +

p

2
− d

y = 2

(
ℜ(λ̃d) +

√
p

2

)
ℑ(λ̃d)− pb

Thus

y2 = 4

(
ℜ(λ̃d) +

√
p

2

)2

ℑ(λ̃d)
2 − 4

(
ℜ(λ̃d) +

√
p

2

)
ℑ(λ̃db)pb+ p2b2

=

[(
ℜ(λ̃d) +

√
p

2

)2

− x+

√
p

2
− d− ipb

]
By setting

p̃d = 2

(
ℜ(λ̃d) +

√
p

2
− pb

)2

we have

y2 = 2p̃db

(
p̃db
2

− x− p

2
+ db

)
= 2p̃db

(
p̃db
2

− x̃

)
We also assume that ∥Adb∥ = 1. Otherwise (if ∥Adb∥ ≠ 1), we can take

A1
db =

1

∥Adb∥
Adb and p̃1db =

1

∥Adb∥
.

Consider the numerical parameters αÃd
and αAdb

defined by

αÃd
= sup

ℜ(λ̃d)=0

∥(λ̃dI2n − Ãd)
−1∥ and αAdb

= sup
z∈Γ̃d

∥(zIn −A)−1∥ (42)

The following proposition gives a relation between the parameters αÃd
and αAdb

.

Proposition 10. Let αÃd
and αAdb

be the two parameters defined in (46). Assume that

∥Adb∥ = 1 and
∣∣∣p
2
− d− ipb

∣∣∣ < 1

αdb
(43)

Then
αAdb

≤ αÃd
≤ 2

(
αAdb

+
√
αAdb

(√
1 + αAdb

+ 1
))

. (44)



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 711

Proof. Let the matrix

(
λ̃I2n − Ãd

)
=


(λ̃d +

√
p
2)In −Adb

−In (λ̃d +
√

p
2)In


We have


(λ̃d +

√
p
2)In Adb

In (λ̃d +
√

p
2)In!

×


(λ̃d +

√
p
2)In −Adb

−In (λ̃d +
√

p
2)In

 =


(λ̃d +

√
p
2)

2In −Adb 0

0 (λ̃d +
√

p
2)

2In −Adb


With

(λ̃dI2n − Ãd)
−1 =


(λ̃d +

√
p
2)

2In −Adb 0

0 (λ̃d +
√

p
2)

2In −Adb


−1

×


(λ̃d +

√
p
2)In Adb

In (λ̃d +
√

p
2)In



=


(λ̃d +

√
p
2)

2In − (A+ (p2 − d− ipb)In) 0

0 (λ̃d +
√

p
2)

2In − (A+ (p2 − d− ipb)In)


−1

×


(λ̃d +

√
p
2)In (A+ (p2 − d− ipb)In)

In (λ̃d +
√

p
2)In



=


((λ̃d +

√
p
2)

2 − p
2 + d− ipb)In −A 0

0 ((λ̃d +
√

p
2)

2 − p
2 + d− ipb)In −A


−1

×



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 712
(λ̃d +

√
p
2)In A+ (p2 − d− ipb)In)

In (λ̃d +
√

p
2)In



=

(zIn −A)−1 0

0 (zIn −A)−1

×


√
z + p

2 − d− ipbIn A+ (p2 − d− ipb)In

In

√
z + p

2 − d− ipbIn



=


√
z + p

2 − d− ipb(zIn −A)−1 (zIn −A)−1(A+ (p2 − d− ipb)In)

(zIn −A)−1
√

z + p
2 − d− ipb(zIn −A)−1


Knowing that the norm of (λ̃dI2n−Ãd)

−1 is greater than or equal to the norm of each
of its block components taken individually, we can deduce that

αÃd
= sup

ℜ(λ̃d)=0

∥(λ̃dI2n − Ãd)
−1∥ ≥ sup

z∈Γ̃d

∥(zIn −A)−1∥ = αAdb

and also

∥∥∥(λ̃dI2n − Ãd)
−1
∥∥∥ ≤

∥∥∥∥∥∥∥∥

√
z + p

2 − d− ipbIn A+ (p2 − d− ipb)In

In

√
z + p

2 − d− ipbIn


∥∥∥∥∥∥∥∥
∥∥(zIn −A)−1

∥∥

≤

∥∥∥∥∥∥∥∥

∥
√
z + p

2 − d− ipbIn∥ ∥A+ (p2 − d− ipb)In∥

∥In∥ ∥
√
z + p

2 − d− ipbIn∥


∥∥∥∥∥∥∥∥
∥∥(zIn −A)−1

∥∥

≤

∥∥∥∥∥∥∥∥

√

|z|+
√
|p2 − d− ipb| 1

1
√

|z|+
√

|p2 − d− ipb|


∥∥∥∥∥∥∥∥
∥∥(zIn −A)−1

∥∥

≤
(√

|z|+
√

|p
2
− d− ipb|+ 1

)∥∥(zIn −A)−1
∥∥



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 713

• If |z| ≤ αAdb
+ 1

αAdb

then

∥(λI2n − Ãd)
−1∥ ≤ αAdb

(
√

|z|+
√
|p
2
− d− ipb|+ 1)

≤ αAdb

(√
αAdb

+ 1

αAdb

+
1√
α
+ 1

)
≤ αAdb

+
√
αdb(

√
αdb + 1 + 1)

• If |z| > αAdb
+ 1

αAdb

with the conditions (43) we have

∥∥∥∥Az
∥∥∥∥ < 1. we note that

∥∥∥∥Az
∥∥∥∥ <

αdb

αdb + 1

(
∥Adb∥+ |p

2
− d− ipb|

)
<

αdb

αdb + 1

(
1 +

1

αdb

)
< 1

Which leads to

(zIn −A)−1 =
1

z
(In − A

z
)−1

=
1

z

(
In +

+∞∑
k=1

Ak

zk

)

=
1

z

(
In +

A

z

+∞∑
m=0

Am

zm

)

=
1

z

(
In +

A

z
(In − A

z
)−1

)
Consequently

∥(λ̃dI2n − Ãd)
−1∥ ≤

∥∥∥∥1z In +
1

z
A (zIn −A)−1

∥∥∥∥× ∥∥∥∥1 +√|z|+
√∣∣∣p

2
− d− ipb

∣∣∣∥∥∥∥

≤ (∥A(zIn −A)−1∥+ 1)
1 +

√
|z|+

√∣∣p
2 − d− ipb

∣∣
|z|



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 714

≤
((

1 + |p
2
− p− ipb|

)
αAdb

+ 1
) 1√

|z|
+

1

|z|
+

√
|p2 − d− ipb|

|z|



≤
((

1 +
1

αAdb

)
+ 1

)( √
αAdb√

αAdb
+ 1

+
αAdb

αAdb
+ 1

+

√
αAdb

αAdb+1

)

≤
(
αAdb

+ 2

αAdb
+ 1

)(√
αAdb

√
αAdb

+ 1 +
√
αAdb

+ αdb

)
≤ 2

(
αAdb

+
√
αAdb

(√
αAdb+1 + 1

))
Finally

αAdb
≤ αAdb

≤ 2
(
αAdb

+
√
αAdb

(√
αAdb+1 + 1

))
.

Consider the projector

• P̃d ∈ Cn×n on right eigenspace of Adb associated with eigenvalues outside the
parabola Γ̃d

• P̃d ∈ C2n×2n on right eigenspace of Ãd associated with eigenvalues in the right com-
plex half-plane.

The following proposition characterizes the relation between P̃d and P̃d

Proposition 11. Consider a partition of the matrix P̃d in the form

P̃d =

(
P̃(d)
1 P̃(d)

2

P̃(d)
3 P̃(d)

4

)
with P̃(d)

i ∈ Cn×n, i = 1, 4 (45)

Then

P̃d = 2P̃(d)
1 = 2P̃(d)

4 = 4P̃(d)
2 P̃(d)

3 (46)

Moreover
P̃dA = 4(P̃(d)

2 )2 − (p− 2d− 2ipb)P̃(d)
1 (47)

Proof.
Let X̃d be a solution of the matrix equation(

X̃d +

√
p

2
In

)2

= Adb. (48)

Following the same calculation as in the proof of Proposition 7, we get



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 715

Ãd =

X̃d +
√

p
2In −X̃d −

√
p
2In

In In

×

X̃d 0

0 −X̃d − 2

√
p

2
In

×


1

2
(X̃d +

√
p

2
In)

−1 1

2
In

−1

2
(X̃d +

√
p

2
In)

−1 1

2
In


Let X̃d = Qdb

[
M+ 0
0 M−

]
Q−1

db be the canonical Jordan form of the matrix X̃d with

M+ and M− the Jordan blocks associated respectively with the eigenvalues of X̃d located
in the right half-plane and the left half-plane.

By replacing the decomposition of X̃d in the matrix Ãd, we get

Ãd = Q̃dbM(Q̃db)
−1

with

Q̃db =

Qdb 0

0 Qdb



(
M+ 0
0 M−

)
+

√
p

2
In −

[
M+ 0
0 M−

]
−
√

p

2
In

In In


et

M =


[
M+ 0
0 M−

]
0

0

[
M+ 0
0 M−

]
− 2

√
p

2
In


We can therefore calculate the associated projector

P̃d = (Q̃db)

[
Ik 0
0 0

]
(Q̃db)

−1

=

Qdb 0

0 Qdb



[
M+ 0
0 M−

]
+

√
p

2
In −

[
M+ 0
0 M−

]
−
√

p

2
In

In In

×


[
Ik 0
0 0

] [
0 0
0 0

]
[
0 0
0 0

] [
0 0
0 0

]


×



1

2

([
M+ 0
0 M−

]
+

√
p

2
In

)−1
1

2
In

−1

2

M+ 0

0 M−

−
√

p

2
In

−1

1

2
In


Q−1

db 0

0 Q−1
db





S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 716

=

Qdb 0

0 Qdb




M+ +

√
p

2
Ik 0

0 0

 [
0 0
0 0

]
[
Ik 0
0 0

] [
0 0
0 0

]



×



1

2

(M+ +

√
p

2
Ik)

−1 0

0 (M− +

√
p

2
In−k)

−1

 1

2

[
Ik 0
0 In−k

]

−1

2

(M+ +

√
p

2
Ik)

−1 0

0 (M− +

√
p

2
In−k)

−1

 1

2

[
Ik 0
0 In−k

]



Q−1
db 0

0 Q−1
db



=

Qdb 0

0 Qdb




1

2

[
Ik 0
0 0

]
1

2

M+ +

√
p

2
Ik 0

0 0


1

2

(M+ +

√
p

2
Ik)

−1 0

0 0

 1

2

[
Ik 0
0 0

]

Q−1

db 0

0 Q−1
db



=


Qdb

[1
2
Ik 0

0 0

]
Q−1

db Qdb

12(M+ +

√
p

2
Ik) 0

0 0

Q−1
db

Qdb

12(M+ +

√
p

2
Ik)

−1 0

0 0

Q−1
db Qdb

[1
2
Ik 0

0 0

]
Q−1

db



=

[
P̃(d)
1 P̃(d)

2

P̃(d)
3 P̃(d)

4

]

It follows that

P̃(d)
1 = Qd

[1
2
Ik 0

0 0

]
Q−1

db =
1

2
P̃d

P̃(d)
2 =

1

2
Qdb

M+ +

√
p

2
Ik 0

0 0

Q−1
db



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 717

P̃(d)
3 =

1

2
Qdb

(M+ +

√
p

2
Ik)

−1 0

0 0

Q−1
db

P̃(d)
4 = Qdb

[1
2
Ik 0

0 0

]
Q−1

db =
1

2
P̃d

With X̃d = Qdb

[
M+ 0
0 M−

]
Q−1

db we have

A = Adb −
(p
2
− d− ipb

)
In

= Qd


(
M+ +

√
p

2
Ik

)2

−
(p
2
− d− ipb

)
Ik 0

0

(
M− +

√
p

2
In−k

)2

−
(p
2
− d− ipb

)
In−k

Q−1

and

P̃dA = Qdb

[
Ik 0
0 0

]
(
M+ +

√
p

2
Ik

)2

− (
p

2
− d− ipb)Ik 0

0

(
M− +

√
p

2
In−k

)2

− (
p

2
− d− ipb)In−k

Q−1
db

= Qdb

(M+ +

√
p

2
Ik

)2

− (
p

2
− d− ipb)Ik 0

0 0

Q−1
db

= 4(P̃(d)
2 )2 − (p− 2d− 2ipb)P̃(db)

1

Hence (46) and (47).

Remark 7. We note that :

• if the parameter d =
p

2
, whence equalities (47) are reduced to those of equalities (31).

• if the parameter b = 0, whence equalities (47) are reduced to those of equalities (40).

• if the parameters b = 0, d =
p

2
, whence equalities (47) are reduced to those of equali-

ties (23).

Algorithm 7 (DichoPdb).



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 718

• Input variables: the matrices A, In and the real numbers b, d and p such that the
matrix pencil zIn −A has no eigenvalues on the parabola with equation 2p (d− x) =
(y − ipb)2 with p > 0 et d > 0

• Output variables: P̃d, H̃d and b ̸= 0.
P̃d being the projector on the right subspace of zIn−A associated with the eigenvalues
outside the parabola and H̃d the matrix whose norm defines the dichotomy criterion.

1. Determine the matrix

Ãd =


−
√

p

2
In A+ (

p

2
− d− ipb)In

In −
√

p

2
In


2. Using Algorithm 3 to λ̃dI2n−Ãd, compute the Projector P̃d onto the right eigenspace

of Ãd associted with the eigenvalues on the right half-plane of the complex plane and
the matrix H̃d.

3. If ∥H̃d∥ is not large, determine the projector P̃d by

P̃d = 2P̃(d)
1 .

4. Numerical experiments

In this section, we illustrate numerical examples using a matrix function from [2, 4, 5]
on which we apply the algorithms 4, 5, 6 and 7 for positive parameters p, b and d given.

W (t) =

−(A(s(t)))−T cos(w(t)) −(A(s(t)))−T sin(w(t))

A(s(t)) sin(w(t)) (A(s(t)))−T cos(w(t))

 (49)

avec

A(s) =

(
1− s2 −1
s2 1− s2

)
, w(t) = π

(
1

2
− 1

3
sin(3t)

)
et s(t) = 4 sin(3t)

• Applying Algorithm 4 to matrix function (49) gives the following results:

· At t = 3.5,

Those different graphs on Figure 1 show how a parabola Γ can realise a di-
chotomy on the eigenvalues of a given matrix. We have three possibilities:
when all the eigenvalues are in the interior,then the computed projector is the
null matrix. When all the eigenvalues are at the exterior of the parabola, the



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 719

−10 0 10
−15

−10

−5

0

5

10

15

The abscissa axis

T
h

e
 o

rd
in

a
te

 a
x
is

p=4 and t=3.5

−10 0 10
−8

−6

−4

−2

0

2

4

6

8

The abscissa axis

T
h

e
 o

rd
in

a
te

 a
x
is

p=1.3  and t=3.5

−10 0 10
−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

The abscissa axis

T
h

e
 o

rd
in

a
te

 a
x
is

p=0.2  and t=3.5

Figure 1: Partition of the spectrum of the matrix W (t) for t = 3.5 by parabolas of equation 2p(
p

2
− x) = y2.

Table 1: Traces, norms and quality of spectral projectors P by applying the DichoP algorithm for three values
of p

p tr(P) ∥P∥ ∥P2 − P∥ ∥PW (t)−W (t)P∥ ∥H∥
4 0 0 0 0 4.0502
1.3 2 1.6305 2.3747 10−15 5.6077 10−15 63.1478
0.2 4 1 2.0540 10−15 5.3639 10−15 6.2228

computed projector is the identity matrix. A part of the eigenvalues can be in
the interior of the parabola and another part of the eigenvalues can be at the ex-
terior. In this case, the projector is different of the null matrix and the identity
matrix. In the above table of values, the trace tr(P) of P denotes the number of
eigenvalues outside of the parabola. Moreover, the values of ∥P∥ confirm what
was said above. The computing of ∥P2 − P∥ and ∥PW (t)−W (t)P∥ prove that
P is a projector and the values obtained for ∥H∥ show the good quality of the
dichotomy. This shows the effectiveness of the method.

· The partition of the eigenvalues of W (t), ∀t ∈ [0, π] with the parameters p ∈
{0.5, 1, 2, 4} gives us Figure 2.

Those graphs describe the spectral portrait of W (t),∀t ∈ [0; 2π]

This spectrum dichotomy realised by the parabola Γ is illustrated with colors
(the green color for the inside eigenvalues and the red color for the outside
eigenvalues).

• Applying Algorithm 5 to matrix function 49 gives the following results:

· At t = 6,

Those different graphs on Figure 3 show how a parabola Γ̃ can realise a di-
chotomy on the eigenvalues of a given matrix. Similarly to the case seen for



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 720

−15 −10 −5 0 5
−4

−2

0

2

4

the abscissa axis
th

e
 o

rd
in

a
te

 a
x
is

p=0.5

−15 −10 −5 0 5
−10

−5

0

5

10

the abscissa axis

th
e

 o
rd

in
a

te
 a

x
is

p=1

−15 −10 −5 0 5
−10

−5

0

5

10

the abscissa axis

th
e

 o
rd

in
a

te
 a

x
is

p=2

−15 −10 −5 0 5
−20

−10

0

10

20

the abscissa axis

th
e

 o
rd

in
a

te
 a

x
is

p=4

Figure 2: Partition of the eigenvalues of W (t), ∀t ∈ [0, π] for p ∈ {0.5, 1, 2, 4}.

Table 2: Traces, norms and quality of spectral projectors P̃ by applying the DichoPb algorithm for differents
values of p and b

p b tr(P̃) ∥P̃∥ ∥P̃2 − P̃∥ ∥P̃W (t)−W (t)P̃∥ ∥H̃∥
2 1 3 1.0268 1.4726 10−15 2.2659 10−15 3.9802
2 3 4 1 2.6170 10−15 3.9255 10−15 0.9799
2 0.1 0 4.9838 10−16 4.9838 10−16 8.0143 10−16 29.4351
0.1 0.5 4 1 2.8478 10−15 5.1651 10−15 1.1260
1 0.5 3 1.6455 2.6392 10−15 2.6392 10−15 7.7339
4 0.5 0 4.8120 10−16 4.8120 10−16 6.0288 10−16 3.7896

the parabola Γ, the values obtained for ∥P̃2− P̃∥ and ∥P̃W (t)−W (t)P̃∥ proved
that P̃ is a projector. Moreover, the values of ∥H̃∥ show the good quality of the
dichotomy.

· The partition of the eigenvalues ofW (t), ∀t ∈ [0, π] with the parameters (p, b) ∈
{(0.5, 2), (1,−2), (2, 0), (4, 3)} gives us Figure 4. This shows the effectiveness
of the method.

Those graphs describe the spectral portrait of W (t),∀t ∈ [0; 2π]. This spectrum
dichotomy realised by the parabola Γ̃ is illustrated with colors (the green color
for the inside eigenvalues and the red color for the outside eigenvalues).

• Applying Algorithm 6 to matrix function 49 gives the following results:

· At t = 2π,

Those different graphs on Figure 5 show how a parabola Γd can realise a di-
chotomy on the eigenvalues of a given matrix.
Similarly to the case seen for the parabola Γ, the values obtained for ∥P2

d−Pd∥



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 721

−15 −10 −5 0 5 10 15
−10

−5

0

5

10

15

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=3 and b=1

−15 −10 −5 0 5 10 15
−5

0

5

10

15

20

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=3 and b=0.2

−15 −10 −5 0 5 10 15
−10

−5

0

5

10

15

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=3 and b=3

−15 −10 −5 0 5 10 15
−10

−5

0

5

10

15

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=4 and b=0.5

−15 −10 −5 0 5 10 15
−6

−4

−2

0

2

4

6

8

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=1 and b=0.5

−15 −10 −5 0 5 10 15
−3

−2

−1

0

1

2

3

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=0.2 and b=0.5

Figure 3: Partition of the spectrum of the matrix W (t) for t = 6 by parabolas of equation 2p(
p

2
−x) = (y−ipb)2.

Table 3: Traces, norms and quality of spectral projectors Pd by applying the DichoPd algorithm for differents
values of p and d

p d tr(Pd) ∥Pd∥ ∥P2
d − Pd∥ ∥PdW (t)−W (t)Pd∥ ∥Hd∥

0.1 5 0 1.9703 10−17 1.9703 10−17 4.1735 10−17 37.0029
0.1 2 4 1 1.3784 10−15 1.9182 10−15 11.0360
0.1 3.5 2 1 2.8954 10−15 5.0227 10−15 96.0484
0.15 1.5 4 1 2.1579 10−15 3.6774 10−15 15.1757
0.25 1.5 2 1 4.4977 10−16 2.0476 10−15 9.6088
2 1.5 0 6.2936 10−17 6.2936 10−17 1.0107 10−16 1.1828

and ∥PdW −WPd∥ proved that Pd is a projector. Moreover, the values of ∥Hd∥
show the good quality of the dichotomy.

· The partition of the eigenvalues ofW (t), ∀t ∈ [0, π] with the parameters (p, d) ∈
{(0.5, 0.5), (2, 1), (0.2, 2), (0.2, 7)} gives us Figure 6. This shows the effectiveness
of the method.

Those graphs describe the spectral portrait of W (t),∀t ∈ [0; 2π] This spectrum
dichotomy realised by the parabola Γd is illustrated with colors (the green color
for the inside eigenvalues and the red color for the outside eigenvalues).

• Applying Algorithm 7 to matrix function 49 gives the following results:

· At t = 3.5,

Those different graphs on Figure 7 show how a parabola Γ̃d can realise a di-
chotomy on the eigenvalues of a given matrix.

Similarly to the case seen for the parabola Γ, the values obtained for ∥P̃d
2
− P̃d∥

and ∥P̃dW (t)−W (t)P̃d∥ proved that P̃d is a projector. Moreover, the values of

∥H̃d∥ show the good quality of the dichotomy.



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 722

−10 0 10
−5

0

5

The abscissa axis
T

h
e

 o
rd

in
a

te
 a

x
is

p=0.5 et b=2

−10 0 10
−5

0

5

10

The abscissa axis

T
h

e
 o

rd
in

a
te

 a
x
is

p=1 et b=−2

−10 0 10
−10

−5

0

5

10

The abscissa axis

T
h

e
 o

rd
in

a
te

 a
x
is

p=3 et b=0

−10 0 10
−30

−20

−10

0

10

The abscissa axis

T
h

e
 o

rd
in

a
te

 a
x
is

p=4 et b=3

Figure 4: Partition of the eigenvalues of W (t), ∀t ∈ [0, π] with (p, b) ∈ {(0.5, 2), (1,−2), (2, 0), (4, 3)}.

−15 −10 −5 0 5
−6

−4

−2

0

2

4

6

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=1 and d=0.6

−15 −10 −5 0 5
−6

−4

−2

0

2

4

6

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=1 and d=0.8

−15 −10 −5 0 5
−6

−4

−2

0

2

4

6

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=1 and d=2

−15 −10 −5 0 5
−4

−3

−2

−1

0

1

2

3

4

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=0.35 and d=1.3

−15 −10 −5 0 5
−10

−5

0

5

10

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=2 and d=1.3

−15 −10 −5 0 5
−3

−2

−1

0

1

2

3

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=0.2 and d=1.3

Figure 5: Partition of the spectrum of the matrix W (t) for t = 2π by parabolas of equation 2p(d− x) = y2.

· The partition of the eigenvalues of W (t), ∀t ∈ [0, π] with the parameters
(p, d, b) ∈ {(0.5, 0.5, 2), (2, 1,−2), (0.2, 2, 0), (0.2, 7, 3)} gives us Figure 8. This
shows the effectiveness of the method.

Those graphs describe the spectral portrait of W (t),∀t ∈ [0; 2π] This spectrum
dichotomy realised by the parabola Γ̃d is illustrated with colors (the green color
for the inside eigenvalues and the red color for the outside eigenvalues).

5. Conclusion

In this work, we proposed methods of spectral dichotomies of a matrix with respect to
the general equation x = ay2 + by+ c wth a ̸= 0 of a parabola Γa,b,c. These methods are
modifications of the Algorithm proposed by A. N. Malyshev and M. Sadkane in [13]. In



S. Traoré, M. Dosso / Eur. J. Pure Appl. Math, 15 (2) (2022), 681-725 723

−15 −10 −5 0 5
−4

−2

0

2

4

l’axe des abscisses
l’a

x
e

 d
e

s
 o

rd
o

n
n

é
e

s

p=0.5 et d=0.5

−15 −10 −5 0 5
−10

−5

0

5

10

l’axe des abscisses

l’a
x
e

 d
e

s
 o

rd
o

n
n

é
e

s

p=2 et d=1

−15 −10 −5 0 5
−5

0

5

l’axe des abscisses

l’a
x
e

 d
e

s
 o

rd
o

n
n

é
e

s

p=0.2 et d=2

−15 −10 −5 0 5
−10

−5

0

5

10

l’axe des abscisses

l’a
x
e

 d
e

s
 o

rd
o

n
n

é
e

s

p=0.2 et d=7

Figure 6: Partition of the eigenvalues of W (t), ∀t ∈ [0, π] with (p, d) ∈ {(0.5, 0.5), (2, 1), (0.2, 2), (0.2, 7)}.

−15 −10 −5 0 5
−6

−4

−2

0

2

4

6

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=1,b=0.2 and d=0.2

−15 −10 −5 0 5
−4

−2

0

2

4

6

8

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=1,b=2 and d=1

−15 −10 −5 0 5
−4

−2

0

2

4

6

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=0.5,b=0.5 and d=2

−15 −10 −5 0 5
−10

−5

0

5

10

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=2,b=0.5 and d=0.3

−15 −10 −5 0 5
−8

−6

−4

−2

0

2

4

6

8

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=1.3,b=0.2 and d=3

−15 −10 −5 0 5
−5

0

5

10

15

abscisses

o
rd

o
n

n
�e

s

for t=3.5 with p=1.8,b=3 and d=3

Figure 7: Partition of the spectrum of the matrixW (t) for t = 2.8 by parabolas of equation 2p(d−x) = (y−pb)2.

this study, if the discriminant is equal to 1 in the canonical form of the general equation
with non-zero parameter b, the matrix A is replaced by the matrix Ab = A− ipbIn in the
algorithm. Moreover, if the discriminant is different from 1, we replace the matrix A by

Ad = A+
(p
2
− d
)
In (respectively Adb = A+

(p
2
− d− ipb

)
In) in the DichoP algorithm

when the parameter b is zero (respectively b is not zero ).
A theoretical analysis of the proposed new methods shows how to calculate the pro-

jector spectral associated with the eigenvalues outside at a given parabola Γ(a, b, c). An
analysis of the new method shows how to extract the projector.

Thus, In the numerical experiments, the application of the four algorithms DichoP,
DichoPb, DichoPd and DichoPdb to a matrix function shows the efficient calculation of
the projector which allows a separation of its spectrum into two parts with respect to the
parabola (i. e. inside and outside the parabola).



REFERENCES 724

Table 4: Traces, norms and quality of spectral projectors P̃d by applying the DichoPdb algorithm for different
values of p b and d

p b d tr(P̃d) ∥P̃d∥ ∥P̃d
2
− P̃d∥ ∥P̃dW (t)−W (t)P̃d∥ ∥H̃d∥

1 0.2 0.2 2 1.3292 3.3183 10−15 7.2032 10−15 7.3681
1 4 1 4 1 3.8213 10−15 5.3470 10−15 1.3453
0.5 0.5 2 0 3.2139 10−17 3.2139 10−17 4.7630 10−16 9.9291
2 0.5 0.3 1 1.3499 1.7537 10−15 5.6241 10−15 4.3035
1.3 0.2 3 0 3.6050 10−17 3.6050 10−17 6.0976 10−17 1.3319
3 5 3 4 1 2.6922 10−15 5.4784 10−15 1.1007

−10 0 10
−5

0

5

the abscissa axis

th
e

 o
rd

in
a

te
 a

x
is

p=0.5, d=0.5 et b=2

−10 0 10
−5

0

5

10

15

the abscissa axis

th
e

 o
rd

in
a

te
 a

x
is

p=2, d=1 et b=−2

−10 0 10
−5

0

5

the abscissa axis

th
e

 o
rd

in
a

te
 a

x
is

p=0.2, d=2 et b=0

−10 0 10
−20

−10

0

10

the abscissa axis

th
e

 o
rd

in
a

te
 a

x
is

p=0.2,  d=7 et b=3

Figure 8: Partition of the eigenvalues of W (t), ∀t ∈ [0, π] with (p, d, b) ∈
{(0.5, 0.5, 2), (2, 1,−2), (0.2, 2, 0), (0.2, 7, 3)}.

Acknowledgements

The authors thank the referees for their useful suggestions that helped to improve this
article.

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