International Journal of Analysis and Applications

Volume 18, Number 6 (2020), 1048-1055

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-1048

GENERALIZED SPECTRUM AND NUMERICAL RANG OF MATRIX THE

LORENTZIAN OSCILLATOR GROUP OF DIMENSION FOUR

RAFIK DERKAOUI∗, ABDERRAHMANE SMAIL

University of Oran 1, Laboratory GEANLAB, P.O. Box 1524, Oran, 31000, Algeria

∗Corresponding author: rafikderkaoui27@yahoo.com

Abstract. In this paper, we find the spectrum, pseudo-spectrum and numerical rang of matrix of the

metric ga.

1. Introduction

Connected Lie groups that admit a bi-invariant Lorentzian metric were determined by the first of the

authors in [14]. Among them, those that are solvable, non-commutative, and simply connected are called

oscillator groups. This group has many properties useful both in geometry and physics.

We study here the geometry of these groups and their networks, i.e their discrete sub-groups co-compact.

If G is an oscillator group, its networks determine compact homogeneous Lorentz manifolds, on which G

acts by isometries.

Let H2k+1 = R × Ck be the Heisenberg group and let λ = (λ1,λ2, . . . ,λk) k be strictly positive real

numbers. Let the additive group R act on H2k+1 by the action:

ρ(t)(u, (zj)) = (u, (e
iλjtzj)).

The group Gk(λ), a semi-direct product of R by H2k+1 following ρ, is provided with a bi-invariant Lorentz

metric. Here is how it is built:

g = R×R×R2k

Received September 18th, 2020; accepted October 9th, 2020; published October 28th, 2020.

2010 Mathematics Subject Classification. 53B30, 47A10, 47A12.

Key words and phrases. oscillator group; spectrum; pseudo-spectrum; numerical rang.

©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

1048

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-1048


Int. J. Anal. Appl. 18 (6) (2020) 1049

is the tangent space at the origin. Let us extend the usual scalar product of R2k into a symmetric bilinear

form over g so that the plane R×R is hyperbolic and orthogonal to R2k. This form defines an invariant

Lorentz metric on the left on Gk(λ), it is also invariant on the right because the adjoint operators on g are

antisymmetric [15].

Groups Gk(λ) are characterized [14] by:

Theorem 1.1. The groups Gk(λ) are the only Lie group simply connected , resolvable and noncommutative

which admit a bi-invariant Lorentz metric.

Remark 1.1. it is easy to see that the groups G1(λ) are isomorphous; the group G1 = G1(1) is usually

known as the oscillator group [20].

Since [1], [2] et [3] the oscillator group has been generalized to a dimension equal to an even number 2n

with n ≥ 2, plus this provides a known example of homogeneous space-time [6].

For n = 2, the oscillator group of dimension 4 admits a Lorentzian metric invariant on the left and on

the right (bi-invariant). This bi-invariant metric has been generalized a family ga, −1 < a < 1, invariant

Lorentzian metrics on the left. For a = 0, the metric g0 become or the only example of Lorentzian bi-invariant

metric [7]

The researchers Giovani and Zaeim extracted three vectors feilds from the oscillator group, which are:

Killing vector feild, Affine vector feild, parallel vector feild (see [4]).

and also Giovani and Zaeim classified the totally geodesic and parallel hypersurfaces of four-dimensional

groups (see [3]).

Varah published an article entitled ”On the separation of two matrices” in which he defined with standard

2 the pseudospectrum using the smallest singular value σmin(zI−A) under the notion Λ�(A) see [23]. In the

1960s the pseudospectrum was studied in several by L. N. Trefethen [19], [21].

In recent years the study of the pseudospectrum has been very active, many contributions related to the

pseudospectrum have been made by various researchers, for example, J. S. Baggett, A. Bottcher, M. Embree,

L. N. Trefethen, L. Reichel, S.C.Reddy, T.A. Driscoll.

The pseudospectrum of a normal matrix A consists of circles of radius � around each eigenvalue. For non-

nirmal matrices, the pseudospectrum takes different forms in the complex plane. in [19] The pseudospectrum

of thirteen highly non-normal matrices is presented.

2. Preliminaries

At the moment we consider on Gλ a family parametre of left-invariant Lorentzian metrics ga. With respect

to coordinates (x1,x2,x3,x4), this metric ga is explicitly given by

ga = adx
2
1 + 2ax3dx1dx2 + (1 + ax

2
3)dx

2
2 + dx

2
3 + 2dx1dx4 + 2x3dx2dx4 + adx

2
4,



Int. J. Anal. Appl. 18 (6) (2020) 1050

with −1 < a < 1.

Note that for a = 0 and λ = 1 we have the bi-invariant metric on the oscillator group G1 [7]. In all other

cases, ga is only invariant on the left.

The matrix of the metric ga is given by

Aa =




a ax3 0 1

ax3 1 + ax
2
3 0 x3

0 0 1 0

1 x3 0 a


 .

Numerical rang

Definition 2.1. Let A be an n×n complex matrix. Then the numerical rang of A, W(A), is defined to be

W(A) =

{
x∗Ax

x∗x
, x ∈ Cn, x 6= 0

}
.

where x∗ denotes the conjugate transpose of the vector x.

Proposition 2.1. Based on the definition of the numerical range, one can now fairly easily deduce the

following basic properties; for details see primarily [ [9], Chapter 1] but also [8].

1− For any A ∈ Mn(C) and for any a,b ∈ C, W(aA + bIn) = aW(A) + b.

2− For any A,B ∈ Mn(C), W(A + B) ⊆ W(A) + W(B).

3− For any A ∈ Mn(C), W(A) contains the convex hull of the eigenvalues of A. If A is normal, i.e.,

A∗A = AA∗, then W(A) equals the convex hull of σ(A).

4− For any A ∈ Mn(C), W(A) ⊂ R if and only if A is Hermitian, i.e., A∗ = A, in this case, the

endpoints of W(A) coincide with the minimum and the maximum eigenvalues of A. Furthermore, W(A) is

a line segment in the complex plane if and only if the matrix A is normal and has collinear eigenvalues; or

equivalently, if and only if A = aH + bI for some a,b ∈ C and an Hermitian matrix H.

3. Eigenvalues and Pseudo-spectrum of matrix Aa

Proposition 3.1. The eigenvalues of the matrix Aa are:

λ1 = 1,

λ2 =
2
3
a + 1

3
ax23 −

1
2
S + C

2S
+ 1

3
−
√
3
2
i
(
S + C

S

)
,

λ3 = λ2,

λ4 =
2
3
a + 1

3
ax23 + S −

C
S

+ 1
3
,

with

S =
3

√
M +

√
N −

8

27
,



Int. J. Anal. Appl. 18 (6) (2020) 1051

and

M =
2

9
a +

1

9
a2 −

1

27
a3 +

1

6
x23 +

11

18
ax23 +

1

6
ax43

−
1

18
a2x23 −

1

18
a3x23 +

1

9
a2x43 +

1

18
a3x43 +

1

27
a3x63

N =
4

27
a3 −

4

27
a2 −

1

27
a4 −

8

27
x23 −

13

108
x43 −

1

27
x63 −

2

9
ax23 −

1

54
ax43 −

1

54
ax63 +

7

27
a2x23

+
4

27
a3x23 +

7

36
a2x43 −

1

9
a4x23 +

1

18
a2x63 −

11

108
a4x43 +

1

27
a3x63 +

1

54
a5x43 −

1

108
a2x83

−
1

108
a6x43 −

1

54
a5x63 +

1

54
a4x83 −

1

54
a6x63 −

1

108
a6x83

C =
2

9
a−

1

9
a2 −

1

3
x23 −

2

9
ax23 −

1

9
a2x23 −

1

9
a2x43 −

4

9

Proof. We have

det(Aa −λI4) = (1 −λ)(−λ3 + Lλ2 + Kλ + (a2 − 1)),

with

L =
(
1 + 2a + ax23

)
,

K = (−a2 − 2a−a2x23 + x
2
3 + 1),

so, det(A−λI4) = 0, If and only if either λ1 = 1 or

−λ3 + Lλ2 + Kλ + (a2 − 1) = 0.

According to the CARDAN method we find,

z3 + pz + q = 0,

such as

(3.1) z = λ−
L

3
, z ∈ C,



Int. J. Anal. Appl. 18 (6) (2020) 1052

and

p = −( 1
3
L2 + K) = −1

3
(4 + a2x43 + ax

2
3 + a

2 − 2a + a2x23 + 3x23) ,

q = − 1
27

(−16 + 2a3x63 + 6a2x43 + 33ax23 − 2a3 + 6a2 − 3a3x23 + 12a + 3a3x43 − 3a2x23 + 9x23) .

Then the CARDAN method he says that the 3 solutions are:

zk = j
k 3

√√√√1
2

(
−q +

√
−∆
27

)
+ j−k 3

√√√√1
2

(
−q −

√
−∆
27

)
, 0 ≤ k ≤ 2

such as ,

∆ = −4p3 − 27q2,

j = ei2
π
3 .

So, according to (3.1) we find,

λk = zk +
L

3
, 0 ≤ k ≤ 2

�

Pseudo-spectrum of Aa: since A is symmetrical therefore Aa is normal, therefore pseudo-spectrum

noted by Λ�(Aa) given by:

Λ�(Aa) = {z ∈ C : |z −λi| ≤ �} with i ∈{1, . . . , 4} .

3.1. Numerical rang of matrix Aa.

Proposition 3.2. The numerical rang of matrix Aa check the following relation:∣∣∣∣x∗Aaxx∗x
∣∣∣∣ ≤ (1 + |a|)(1 + |x3|) + ∣∣ax23∣∣

Proof. We have

W(A) =

{
x∗Ax

x∗x
, x ∈ C4, x 6= 0

}

we put x =




z1

z2

z3

z4


 , with zi = rie

iθi. We have

x∗Aax = a |z1|
2

+ a |z4|
2

+ |z2|
2

+ |z3|
2

+ ax3(z1z2 + z2z1) + x3(z2z4 + z4z2) + (z1z4 + z4z1) + a |z2|
2
x23,



Int. J. Anal. Appl. 18 (6) (2020) 1053

so,

x∗Ax

x∗x
= 1 +

(a− 1)(|z1|
2

+ |z4|
2
)

4∑
i=1

|zi|
2

+ ax3
z1z2 + z2z1

4∑
i=1

|zi|
2

+ x3
z2z4 + z4z2

4∑
i=1

|zi|
2

+
z1z4 + z4z1

4∑
i=1

|zi|
2

+ ax23
|z2|

2

4∑
i=1

|zi|
2

.

We have

(3.2)
|zj|

2

4∑
i=1

|zi|
2

≤ 1, ∀j ∈{1, . . . , 4} .

and

(3.3)
zizj + zjzi

4∑
i=1

|zi|
2

≤ 1, ∀i,j ∈{1, . . . , 4} ,

So from (3.2) and (3.3) we find∣∣∣∣x∗Aaxx∗x
∣∣∣∣ ≤ 1 + |ax3| + |x3| + |a| + ∣∣ax23∣∣ .

It had to be proven. �

Example 3.1. 1) For a = 0 and x3 = 0,

A00 =




0 0 0 1

0 1 0 0

0 0 1 0

1 0 0 0


 ,

so

g00(x
∗,x)

x∗x
= 1 −

r21 + r
2
4 − 2r1r4 cos(θ1 −θ4)
r21 + r

2
2 + r

2
3 + r

2
4

≤ 1,

moreover 1 ∈ W(A00)

On the other hand, we have

−
r21 + r

2
4 − 2r1r4 cos(θ1 −θ4)
r21 + r

2
2 + r

2
3 + r

2
4

≥−2,

therefore

g00(x
∗,x)

x∗x
≥−1,

moreover −1 ∈ W(A00). So W(A00) = [−1, 1]

2) For a = 0 and x3 = 0.5,

A0.50 =




0 0 0 1

0 1 0 0.5

0 0 1 0

1 0.5 0 0


 ,



Int. J. Anal. Appl. 18 (6) (2020) 1054

so

g0.50 (x
∗,x)

x∗x
=
g00(x

∗,x)

x∗x
+
r2r4 cos(θ2 −θ4)
r21 + r

2
2 + r

2
3 + r

2
4

.

We have,

r2r4 cos(θ2 −θ4)
r21 + r

2
2 + r

2
3 + r

2
4

≤
1

2
,

and

g0.50 (x
∗,x)

x∗x
≥−

5

4

so

−
5

4
≤
g0.50 (x

∗,x)

x∗x
≤

3

2
,

but −5
4

and 3
2

does not belong to W(A0.50 ), so we get W(A
0.5
0 ) ⊂

]
−5

4
, 3
2

[
.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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	1. Introduction
	2. Preliminaries
	3. Eigenvalues and Pseudo-spectrum of matrix Aa
	3.1. Numerical rang of matrix Aa

	References