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"Misalkan graf G adalah sebuah graf sederhana tak berarah dengan himpunan simpul V dan himpunan busur E, di mana 𝑛=|𝑉| dan 𝑚=|𝐺| berturut-turut menyatakan banyaknya simpul dan busur graf G. Pelabelan graceful adalah suatu pemetaan injektif f yang memetakan himpunan simpul ke {0,1,2…m} yang menginduksi pemetaan bijektif 𝜆 yang memetakan himpunan busur ke {1,2,…m}, dimana label busur tersebut merupakan selisih dari label simpul yang dihubungkan oleh busur tersebut. Graf yang mempunyai pelabelan graceful disebut graf graceful.Untuk graf G dengan m busur dan pemetaan 𝑓:𝑉(𝐺)→ 0,1,2,…𝑚 maka matriks adjacency diperumum adalah matriks 𝐴 𝑚+1 ×(𝑚+1) dengan entri 𝑎𝑖𝑗 adalah 1 apabila terdepat busur vivj yang menghubungkan simpul vi berlabel i dan simpul vj berlabel j. Matriks adjacency diperumum akan digunakan untuk mengkonstruksi graf graceful baru dari graf yang telah diketahui graceful. Konstruksi dilakukan dengan tiga cara. Pertama adalah dengan pemindahan entri matriks adjacency. Kedua adalah dengan pengabungan matriks adjacency dan penggantian entri diagonal tertentu. Ketiga adalah penggabungan matriks adjacency dan penambahan baris dan kolom. Hasil lain yang diperoleh adalah kelas graf graceful baru: 𝑃𝑝△𝐶𝑛 dan 𝐾1⋄𝑝𝐺."

"Starting with matrices over the integers, part 1 of this book provides a measured introduction to such groups, two finitely generated abelian groups are isomorphic if and only if their invariant factor sequences are identical. The analogous theory of matrix similarity over a field is then developed in part 2 starting with matrices having polynomial entries, two matrices over a field are similar if and only if their rational canonical forms are equal. Under certain conditions each matrix is similar to a diagonal or nearly diagonal matrix, namely its Jordan form."

"Misalkan G adalah suatu graf dengan V(G) yang merupakan himpunan simpul tak kosong dan E(G) yang merupakan himpunan busur. Hubungan tetangga antar simpul dalam suatu graf dapat direpresentasikan dalam bentuk matriks yang disebut matriks adjacency, dengan entrinya bernilai 1 apabila terdapat busur di antara dua simpul dan bernilai 0 untuk lainnya. Jika A adalah matriks adjacency dari graf berarah G, maka dapat dibentuk suatu det(xA+I). Pada skripsi ini dijelaskan representasi bentuk det(XA+I) dengan A merupakan matriks adjacency dari graf berarah sederhana.

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Let G be a graph with V(G) is a nonempty set of vertices and E(G) is a set of arcs. A graph can be representated by a matrix called adjacency matrix, with its entry equal to 1 if there is an edge between two vertices in and equal to 0 for others. If A is the adjacency matrix of a directed graph , it can be formed det(xA+I). In this Skripsi is given a representation of det(xA+I) with A is an adjacency matrix of simple directed graph."

"ABSTRAKSkripsi ini dilatarbelakangi oleh permasalahan 49-4 yang dikemukakan oleh Rajesh Pereira mengenai Teorema Perron-Frobenius pada relasi urutan total dalam buletin International Linear Algebra Society, yaitu "IMAGE". Pada skripsi ini dibahas pembuktian pernyataan sebab-akibat dari Teorema Perron-Frobenius yang menyatakan bahwa jika ada sebuah urutan total ≥ pada yang diawetkan (preserved) oleh A sedemikian sehingga ( ,≥) merupakan ruang vektor terurut, maka seluruh nilai eigen dari A adalah bilangan riil non-negatif.

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ABSTRACTThis paper was motivated by problem 49-4 from International Linear Algebra Society buletin, "IMAGE" which was proposed by Rajesh Pereira about Perron-Frobenius theorem for total orders. This paper presented the proof of Perron-Frobenius theorem which state that if there exists a total order ≥ on which is preserved by A and which makes (,≥) an ordered vector space, then all the eigenvalues of A are real and nonnegative."

"Long considered to be a classic in its field, this was the first book in English to include three basic fields of the analysis of matrices -- symmetric matrices and quadratic forms, matrices and differential equations, and positive matrices and their use in probability theory and mathematical economics. Written in lucid, concise terms, this volume covers all the key aspects of matrix analysis and presents a variety of fundamental methods. Originally published in 1970, this book replaces the first edition previously published by SIAM in the Classics series. Here you will find a basic guide to operations with matrices and the theory of symmetric matrices, plus an understanding of general square matrices, origins of Markov matrices and non-negative matrices in general, minimum- maximum characterization of characteristic roots, Krnoecker products, functions of matrices, and much more. These ideas and methods will serve as powerful analytical tools. In addition, this volume includes exercises of all levels of difficulty and many references to original papers containing further results. The problem sections contain many useful and interesting results that are not easily found elsewhere. A discussion of the theoretical treatment of matrices in the computational solution of ordinary and partial differential equations, as well as important chapters on dynamic programming and stochastic matrices are also included."

"Matrix functions are of growing interest due to their fascinating theory and the many applications in which they provide insight and succinct solutions. Functions of Matrices: Theory and Computation gives a thorough treatment of the theory of matrix functions and numerical methods for computing them, as well as an overview of applications."

"Perturbation Bounds for Matrix Eigenvalues contains a unified exposition of spectral variation inequalities for matrices. The text provides a complete and self-contained collection of bounds for the distance between the eigenvalues of two matrices, which could be arbitrary or restricted to special classes. The book emphasis on sharp estimates, general principles, elegant methods, and powerful techniques, makes it a good reference for researchers and students."

"Computational scientists often encounter problems requiring the solution of sparse systems of linear equations. Attacking these problems efficiently requires an in-depth knowledge of the underlying theory, algorithms, and data structures found in sparse matrix software libraries. Here, Davis presents the fundamentals of sparse matrix algorithms to provide the requisite background. The book includes CSparse, a concise downloadable sparse matrix package that illustrates the algorithms and theorems presented in the book and equips readers with the tools necessary to understand larger and more complex software packages.With a strong emphasis on MATLAB and the C programming language, Direct Methods for Sparse Linear Systems equips readers with the working knowledge required to use sparse solver packages and write code to interface applications to those packages. The book also explains how MATLAB performs its sparse matrix computations."