شماره راهنما :
1922 دكتري
پديد آورنده :
شكوه سلجوقي، حجر
عنوان :
جهش ها و كاربرد آنها در جبر ماكس و جبر خطي
محل تحصيل :
اصفهان : دانشگاه صنعتي اصفهان
صفحه شمار :
[هشت]،[186]ص.: مصور
استاد راهنما :
محمود منجگاني، آلجوسا پپركو
توصيفگر ها :
جبر ماكس , مقادير ويژه ماكس , بردارهاي ويژه ماكس , جهش ها , مقادير ويژه متمايز شده , سري هاي تواني ماكس , بردار ويژه مشترك ماكس , ماتريس هاي جونز , ماتريس هاي دلا پنته
استاد داور :
عباس سالمي، حميدرضا افشين، بهناز عمومي
تاريخ ورود اطلاعات :
1401/04/04
تاريخ ويرايش اطلاعات :
1401/04/04
چكيده فارسي :
هدف اصلي اين رساله مطالعه و بررسي خواص طيفي ماتريس ها در جبرماكس است. براي اين
منظور به كمك مفهومي تحت عنوان جهش به بيان يك الگوريتم جديد به نام الگوريتم جهشي به محاسبه طيف و
فضاي ويژه پرداخته شده است. از جمله مزيت هاي اين روش نسبت به روش هاي قبلي اين است كه زمان رسيدن
به جواب را كوتاه تر مي كند. همچنين به كمك اين مفهوم برخي از ويژگي هاي ماتريس هاي تحويل ناپذير نامنفي
در جبر خطي معمولي بيان شده است. در ادامه رساله نامساوي هاي ماتريسي كه شامل مقايسه توان هاي طيفي
هادامارد و توان هاي طيفي ماكس است مورد بررسي قرار گرفته است. بعلاوه سري هاي تواني در جبر ماكس
بيان شده و خواص طيفي آنها مورد مطالعه قرار گرفته است. همچنين در اين رساله به بيان و بررسي مقادير
ويژه متمايز شده در جبر خطي پرداخته شده و قضيه نگاشت طيفي براي آنها بيان شده است. در پايان توجه ما
به مطالعه خواص طيفي ماتريس هاي جابه جايي در جبر ماكس معطوف گشته است.
چكيده انگليسي :
In this thesis, the spectral properties of non-negative matrices in max algebra have been
studied. It is motivated by the Perron-Frobenius theory as a powerful tool. Throughout
the thesis, we consider max-algebraic versions of some standard results of non-negative
matrices in linear algebra. We have extended many that many well-known theorems in
linear algebra to the max algebra. We start by describing fundamental concepts that will
be used throughout the thesis after a general introduction.
The overview of this thesis is as follows.
In Chapter 1, we provide essential mathematical background on non-negative linear
algebra. Most of the notations used in this thesis are consistent with the literature, each
new symbol is explained on the page where it is introduced, and in order to prevent
confusion some notations and definitions are repeated in each chapter that makes use of
them. We also state well-known results in linear algebra for non-negative matrices. We
are specifically interested in important spectral properties of non-negative matrices.
In Chapter 2, we have introduced max algebra. We have also studied the spectral properties of matrices in max algebra. We have studied a large number of methods for obtaining the spectrum of matrices in max algebra. We define the eigenvalue problem and
recall the Perron-Frobenius theory in each case. Also, we studied numerical approaches
to compute the eigenvalues and eigenvectors of a non-negative irreducible matrix. We
concentrate on linear-algebraic aspects, presenting both classical and new results. Most
of the theory is illustrated by numerical examples and complemented by exercises.
In Chapter 3, we introduce a new method, which is called the mutation method. This
new concept arises in the definition of determinant and is used for calculating max eigencone of non-negative n × n matrix A. The advantage of using the mutation method lies
in shortening a long process of finding the answer or in cutting down the expenditure
by means of simple and clear-cut method. We also present some ways of recognizing
the reducibility and irreducibility of matrices. Some instructive examples are indicated.
In Chapter 4, we are also interested in the characterization of the spectral inequalities
in max algebra. We are specifically interested in the spectral and distinguished eigenvalues of non-negative matrices. We prove new explicit asymptotic formulae between
(geometric) eigenvalues in max-algebra and classical distinguished eigenvalues of nonnegative matrices, is useful tool for transferring results between both settings. We establish new inequalities for both types of eigenvalues of Hadamard products and Hadamard
weighted geometric means of non-negative matrices. Moreover, a version of the spectral mapping theorem for the distinguished spectrum is pointed out.
In Chapter 5, we then focus to find common eigenvectors corresponding to rej
for commuting matrices A and B in max algebra. We study properties of Jones matrix and De
la Puente matrices, some relevant example are indicated.
استاد راهنما :
محمود منجگاني، آلجوسا پپركو
استاد داور :
عباس سالمي، حميدرضا افشين، بهناز عمومي
