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All Journal Journal of Natural Sciences and Mathematics Research UNEJ e-Proceeding Jurnal Matematika Sains dan Teknologi Jurnal Pendidikan Matematika (JUDIKA EDUCATION) Teorema: Teori dan Riset Matematika M A T H L I N E : Jurnal Matematika dan Pendidikan Matematika Vygotsky: Jurnal Pendidikan Matematika dan Matematika Jurnal Ilmiah Matematika dan Pendidikan Matematika (JMP) Euler : Jurnal Ilmiah Matematika, Sains dan Teknologi Jurnal Riset Matematika Joong-Ki Joong-Ki International Journal of Mathematics, Statistics, and Computing
Suroto Suroto
Departement Of Mathematics, Universitas Jenderal Soedirman
Religion Agriculture, Biological Sciences & Forestry Humanities Biochemistry, Genetics & Molecular Biology Chemistry Computer Science & IT Education Environmental Science Materials Science & Nanotechnology Mathematics Physics Social Sciences Other

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Radius Spektral pada Matriks Leslie Novi Andayani Triana; Suroto Suroto; Najmah Istikaanah
Jurnal Riset Matematika Volume 6, No.1, Juli 2026, Jurnal Riset Matematika (JRM)
Publisher : UPT Publikasi Ilmiah Unisba

Show Abstract | Download Original | Original Source | Check in Google Scholar

Abstract

Abstract. One of the matrices commonly used in the field of demography is the Leslie matrix. The Leslie matrix is a matrix that can provide information about the age distribution of the population, especially for women. This study discusses the eigenvalues, spektral radius, the properties of spektral radius related to the power of Leslie matrix. The eigenvalues are determined using the characteristic equation. The determination of spektral radius of the Leslie matrix uses the dominant eigenvalues obtained. Meanwhile, the determination of the properties of the spektral radius of the Leslie matrix is based on properties of the power of eigenvalues of Leslie matrix. The results show that the eigenvalue criteria in Leslie matrix are zero if entry in the first row of the last column is zero. Furthermore, a positive eigenvalue in Leslie matrix has a single existence and is a dominant eigenvalue. Based on dominant eigenvalue, it is known that spektral radius of Leslie matrix will be the same as dominant eigenvalue. Furthermore, the research results also show that spektral radius of the power of a Leslie matrix has same value as the power of its spektral radius, the convergence towards zero of power of a Leslie matrix results in spektral radius of Leslie matrix being less than 1, and the spektral radius of the power of a Leslie matrix being less than or equal to the norm of power of Leslie matrix. Abstrak. Salah satu matriks yang umum digunakan dalam bidang demografi ialah matriks Leslie. Matriks Leslie merupakan matriks yang dapat memberikan informasi mengenai distribusi usia penduduk, khususnya yang berjenis kelamin perempuan. Artikel ini membahas nilai eigen dan radius spektral matriks Leslie serta sifat-sifat radius spektral yang berkaitan dengan hasil pangkat matriks Leslie. Adapun nilai eigen ditentukan menggunakan persamaan karakteristik matriks Leslie. Penentuan radius spektral matriks Leslie dilakukan dengan menggunakan nilai eigen dominan yang diperoleh. Sementara itu, penentuan sifat-sifat radius spektral matriks Leslie dilakukan berdasarkan sifat hasil pangkat nilai eigen dari matriks Leslie. Hasil penelitian menunjukkan bahwa kriteria nilai eigen pada matriks Leslie yaitu bernilai nol jika entri pada baris pertama kolom terakhirnya bernilai nol. Selanjutnya, nilai eigen yang bernilai positif pada matriks Leslie memiliki eksistensi yang tunggal dan merupakan suatu nilai eigen dominan. Berdasarkan nilai eigen dominan tersebut, diketahui bahwa radius spektral matriks Leslie akan bernilai sama dengan nilai eigen dominan tersebut. Selanjutnya, hasil penelitian juga menunjukkan bahwa radius spektral dari hasil pangkat suatu matriks Leslie bernilai sama dengan hasil pangkat dari radius spektralnya, kekonvergenan menuju nol dari hasil pangkat suatu matriks Leslie mengakibatkan nilai radius spektral matriks Leslie tersebut kurang dari 1, serta radius spektral dari hasil pangkat suatu matriks Leslie bernilai kurang dari atau sama dengan norm hasil pangkat matriks Leslie tersebut.
CHARACTERIZATION OF NORMAL FUZZY SUBGROUP Bagas Dwi Cahyo; Suroto Suroto; Sri Maryani
Mathline : Jurnal Matematika dan Pendidikan Matematika Vol. 10 No. 2 (2025): Mathline : Jurnal Matematika dan Pendidikan Matematika
Publisher : Universitas Wiralodra

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31943/mathline.v10i2.791

Abstract

In this article, we discuss the characterization of a normal fuzzy subgroup of classical group . The discussion of this characterization is carried out using Abelian properties, fuzzy conjugate subgroups, fuzzy normalizers, ????-level sets, and fuzzy cosets. The result shows that a sufficient and necessary condition for a normal fuzzy subgroup is the fulfilment of the Abelian condition in the fuzzy subgroup. Then, the equality between of the membership value of all element of  and its conjugate elements is also a sufficient and necessary condition for normal fuzzy subgroup of ????. Moreover, sufficient and necessary conditions of the normal fuzzy subgroup are the normalizer of this fuzzy subgroup is equal to ????. Henceforth, the sufficient and necessary conditions of a normal fuzzy subgroup of  is its ????-level set is a normal subgroup of . Meanwhile, the similarity of the fuzzy right coset and fuzzy left coset of the fuzzy subgroup is also a sufficient and necessary condition for the normal fuzzy subgroup. Furthermore, the normal properties of subgroups on classical groups are a special case of the normal properties in fuzzy subgroups.
Co-Authors Achmad Abdurrazzaq Agung Prabowo Agung Prabowo agus sugandha Agustini Tripena Ari Wardayani Ari Wardayani, Ari Ari Wardayanti Bagas Dwi Cahyo Bagas Sadatmaja Suyatno Putra Chumaedi Sugihandardji Doberhol, James Fajar Hoeruddin Fatkhur Rozi Guswanto, Bambang Hendriya IB Putra Jandhana Ira Lefiana Mashuri Mashuri Melinasari, Wiwit Mugi Lestari, Mugi Muntafi’ah, Naailah Nadya Novika Najmah Istikaanah Najmah Istikaanah Najmah Istikaanah Niken Larasati Novi Andayani Triana Siti Rahmah Nurshiami, Siti Rahmah slamet riyadi Sri Maryani Sri Maryani Sulistyadi, Edy Wardayanti, Ari