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A COMPUTATION PERSPECTIVE FOR THE EIGENVALUES OF CIRCULANT MATRICES INVOLVING GEOMETRIC PROGRESSION SISWANDI SISWANDI; SUGI GURITMAN; NUR ALIATININGTYAS; TEDUH WULANDARI
Jurnal Matematika UNAND Vol 12, No 1 (2023)
Publisher : Departemen Matematika dan Sains Data FMIPA Universitas Andalas Padang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.25077/jmua.12.1.65-77.2023

In this article, the eigenvalues and inverse of circulant matrices with entries in the first row having the form of a geometric sequence are formulated explicitly in a simple form in one theorem. The method for deriving the formulation of the determinant and inverse is simply using elementary row or column operations. For the eigenvalues, the known formulation of the previous results is simplified by considering the specialty of the sequence and using cyclic group properties of unit circles in the complex plane. Then, the algorithm of eigenvalues formulation is constructed, and it shows as a better computation method.

A FAST COMPUTATION FOR EIGENVALUES OF CIRCULANT MATRICES WITH ARITHMETIC SEQUENCE Sugi Guritman; Jaharuddin; Teduh Wulandari Mas'oed; Siswandi
MILANG Journal of Mathematics and Its Applications Vol. 19 No. 1 (2023): MILANG Journal of Mathematics and Its Applications
Publisher : Dept. of Mathematics, IPB University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.29244/milang.19.1.69-80

In this article, we derive simple formulations of the eigenvalues, determinants, and also the inverse of circulant matrices whose entries in the first row form an arithmetic sequence. The formulation of the determinant and inverse is based on elementary row and column operations transforming the matrix to an equivalent diagonal matrix so that the formulation is obtained easily. Meanwhile, for the eigenvalues formulation, we simplify the known result of formulation for the general circulant matrices by exploiting the properties of the cyclic group induced by the set of all roots of as the set of points in the unit circle in the complex plane, and also by considering the specific property of arithmetic sequence. Then, we construct an algorithm for the eigenvalues formulation. This algorithm shows a better computation compared to the previously known result for the general case of circulant matrices.

DETERMINAN, INVERS, DAN NILAI EIGEN MATRIKS SKEW-CIRCULANT DENGAN ENTRI BARISAN GEOMETRI Mirza Farhan Azhari; Teduh Wulandari Mas'oed; Sugi Guritman; Jaharuddin; Siswandi
MILANG Journal of Mathematics and Its Applications Vol. 19 No. 2 (2023): MILANG Journal of Mathematics and Its Applications
Publisher : Dept. of Mathematics, IPB University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.29244/milang.19.2.129-140

Matriks skew-circulant adalah matriks segi yang entri terakhir setiap baris berpindah ke posisi utama dan berganti tanda disertai pergeseran semua entri lainnya ke posisi berikutnya. Dalam artikel ini, entri dari matriks circulant berupa entri barisan bilangan geometri. Tujuannya adalah merumuskan suatu formulasi sederhana dari determinan, invers, dan nilai eigen dari suatu matriks skew circulant. Formulasi determinan ditentukan dengan menerapkan serangkaian operasi baris dasar dan kolom dasar sampai diperoleh matriks diagonal. Langkah untuk mencari invers dilakukan dengan mengadaptasi metode dalam mencari determinan dan ekuivalensi baris dan kolom pada matriks. Dalam mencari nilai eigen digunakan konsep akar kesatuan (roots of unity) dan subgrup siklik.

A FAST COMPUTATION FOR EIGENVALUES OF CIRCULANT MATRICES WITH ARITHMETIC SEQUENCE Guritman, Sugi; Jaharuddin; Mas'oed, Teduh Wulandari; Siswandi
MILANG Journal of Mathematics and Its Applications Vol. 19 No. 1 (2023): MILANG Journal of Mathematics and Its Applications
Publisher : School of Data Science, Mathematics and Informatics, IPB University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.29244/milang.19.1.69-80

In this article, we derive simple formulations of the eigenvalues, determinants, and also the inverse of circulant matrices whose entries in the first row form an arithmetic sequence. The formulation of the determinant and inverse is based on elementary row and column operations transforming the matrix to an equivalent diagonal matrix so that the formulation is obtained easily. Meanwhile, for the eigenvalues formulation, we simplify the known result of formulation for the general circulant matrices by exploiting the properties of the cyclic group induced by the set of all roots of as the set of points in the unit circle in the complex plane, and also by considering the specific property of arithmetic sequence. Then, we construct an algorithm for the eigenvalues formulation. This algorithm shows a better computation compared to the previously known result for the general case of circulant matrices.

DETERMINAN, INVERS, DAN NILAI EIGEN MATRIKS SKEW-CIRCULANT DENGAN ENTRI BARISAN GEOMETRI Azhari, Mirza Farhan; Wulandari Mas'oed, Teduh; Guritman, Sugi; Jaharuddin; Siswandi
MILANG Journal of Mathematics and Its Applications Vol. 19 No. 2 (2023): MILANG Journal of Mathematics and Its Applications
Publisher : School of Data Science, Mathematics and Informatics, IPB University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.29244/milang.19.2.129-140

Matriks skew-circulant adalah matriks segi yang entri terakhir setiap baris berpindah ke posisi utama dan berganti tanda disertai pergeseran semua entri lainnya ke posisi berikutnya. Dalam artikel ini, entri dari matriks circulant berupa entri barisan bilangan geometri. Tujuannya adalah merumuskan suatu formulasi sederhana dari determinan, invers, dan nilai eigen dari suatu matriks skew circulant. Formulasi determinan ditentukan dengan menerapkan serangkaian operasi baris dasar dan kolom dasar sampai diperoleh matriks diagonal. Langkah untuk mencari invers dilakukan dengan mengadaptasi metode dalam mencari determinan dan ekuivalensi baris dan kolom pada matriks. Dalam mencari nilai eigen digunakan konsep akar kesatuan (roots of unity) dan subgrup siklik.

BILANGAN KETERHUBUNGAN PELANGI PADA GRAF GERIGI Diah Prastiwi; Fendy Septyanto; Sugi Guritman; Teduh Wulandari; Siswandi
MILANG Journal of Mathematics and Its Applications Vol. 22 No. 1 (2026): MILANG Journal of Mathematics and Its Applications
Publisher : School of Data Science, Mathematics and Informatics, IPB University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.29244/milang.22.1.27-34

Pewarnaan pelangi pada graf G adalah suatu pelabelan sisi dengan sifat bahwa setiap pasang simpul dapat dihubungkan oleh lintasan pelangi (lintasan yang warna/label sisinya berbeda semua). Bilangan keterhubungan pelangi rc(G) adalah banyaknya warna paling sedikit pada pewarnaan pelangi di graf G. Graf gerigi (gear) G_n diperoleh dari graf roda (wheel) W_n dengan menyisipkan satu simpul pada setiap sisi lingkaran luar. Penelitian sebelumnya menyelidiki rc(G_n) untuk 2<=n<=8, sedangkan kasus n>=9 belum tertangani. Penelitian ini menutup celah tersebut dengan menunjukkan rc(G_n)=min{4, n} untuk n>=2.

Explicit Determinant and Inverse Formulas of Skew Circulant Matrices with Alternating Fibonacci Numbers Sapto Mukti Handoyo; Sugi Guritman; Teduh Wulandari Mas'oed; Jaharuddin Jaharuddin
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 10, No 2 (2025): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v10i2.32358

Skew circulant matrices have various applications such as cryptography, signal processing, and many more. Their structure can potentially simplify their determinant and inverse computations. This study presents explicit formulas for the determinant and inverse of skew circulant matrices with entries from the alternating Fibonacci sequence. Elementary row and column operations are used to derive simple explicit formulas for the determinant and inverse. Computational tests using Wolfram Mathematica show that the algorithm built from these explicit formulas performs with much faster execution time than the built-in functions, especially for large matrix size. The proposed approach offers a practical method for the numerical computation of the determinant and inverse of these matrices

Cheating Detection and Identification in Shamir Secret Sharing Scheme using Parity-Based Verification Mirza Farhan Azhari; Sugi Guritman; Jaharuddin Jaharuddin; Teduh Wulandari Mas'oed
ZERO: Jurnal Sains, Matematika dan Terapan Vol 10, No 2 (2026): Zero: Jurnal Sains Matematika dan Terapan
Publisher : UIN Sumatera Utara

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30829/zero.v10i2.28985

Shamir's (k, n)-threshold secret sharing scheme provides information-theoretic secrecy against unauthorized subsets. However, it does not verify whether submitted shares are genuine during reconstruction. This study proposes a parity-based modification of Shamir's scheme. In the proposed construction, each share is extended into a triplet over Zp (the integers modulo p). The additional component ci serves as an authenticated verification parameter bound to the participant identity and share value. Under the authenticated parity model, the proposed scheme detects and identifies forged shares before reconstruction. Once a forged share is identified, its original value can be restored locally from the authenticated parity value. The cheating success probability is bounded by 1/p. The construction also attains the Ogata-Kurosawa-Stinson lower bound on share size and preserves the threshold reconstruction property. The secret is embedded as the leading coefficient ak-1 and recovered using a recursive divided-difference formulation, requiring O(k2) field operations and O(k) memory after verification. Runtime evaluation over a 256-bit prime field shows that reconstruction remains practical for large reconstruction sets. The additional parity component yields information rate rho = 1/2. It also requires only one extra field element per participant. A blockchain wallet key-distribution case study is included to illustrate how authenticated parity values can support share verification and correction in institutional key recovery. Compared with prior cheating-detection schemes, the proposed construction achieves an OKS-bound cheating probability within a Shamir-based framework. It also supports recursive coefficient recovery and post-identification share correction.

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