GENERALIZED SPECTRUM AND NUMERICAL RANG OF MATRIX THE LORENTZIAN OSCILLATOR GROUP OF DIMENSION FOUR

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

is the tangent space at the origin. Let us extend the usual scalar product of R 2k into a symmetric bilinear form over g so that the plane R × R is hyperbolic and orthogonal to R 2k . This form defines an invariant Lorentz metric on the left on G k (λ), it is also invariant on the right because the adjoint operators on g are antisymmetric [15].
Groups G k (λ) are characterized [14] by: Theorem 1.1. The groups G k (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 G 1 (λ) are isomorphous; the group G 1 = G 1 (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 g a , −1 < a < 1, invariant Lorentzian metrics on the left. For a = 0, the metric g 0 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 σ min (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 The pseudospectrum of a normal matrix A consists of circles of radius around each eigenvalue. For nonnirmal matrices, the pseudospectrum takes different forms in the complex plane. in [19] The pseudospectrum of thirteen highly non-normal matrices is presented.

Preliminaries
At the moment we consider on G λ a family parametre of left-invariant Lorentzian metrics g a . With respect to coordinates (x 1 , x 2 , x 3 , x 4 ), this metric g a is explicitly given by Note that for a = 0 and λ = 1 we have the bi-invariant metric on the oscillator group G 1 [7]. In all other cases, g a is only invariant on the left.
The matrix of the metric g a is given by 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 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 ∈ M n (C) and for any a, b ∈ C, W (aA + bI n ) = aW (A) + b.

Eigenvalues and Pseudo-spectrum of matrix A a
Proposition 3.1. The eigenvalues of the matrix A a are:

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Proof. We have According to the CARDAN method we find, Then the CARDAN method he says that the 3 solutions are: such as , So, according to (3.1) we find, Pseudo-spectrum of A a : since A is symmetrical therefore A a is normal, therefore pseudo-spectrum noted by Λ (A a ) given by: 3.1. Numerical rang of matrix A a .

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