# Matrix Analysis (MATH333) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Matrix Analysis MATH333 3 0 0 3 6
Pre-requisite Course(s)
Math 231 Lineer Cebir I
Course Language English N/A Bachelor’s Degree (First Cycle) Face To Face Lecture, Question and Answer, Drill and Practice. Linear algebra and matrix theory have been fundamental tools in mathematical disciplines. Having the basic knowlegde and properties of linear transformations, vector spaces, vectors and matrices the aim is to present classical and recent results of matrix analysis that have proved to be important to applied mathematics. The students who succeeded in this course; understand Gersgorin’s Circle Theorem and related theorems, and use them, determine whether a given family is simultaneously diagonalizable or not, understand Schur’s Theorem and use it for the triangularization of a matrix, understand Normal matrices, Hermitian matrices and their properties, and perform the QR factorization, triangular factorizations, LU decompositions, determine the minimal polynomials and possible canonical forms of matrices, understand vector and matrix norms, and their properties and using matrix norms determine bounds for the spectral radiuses of matrices, understand the effect of perturbations in the solution of systems of linear equations, understand the Singular Value Decomposition and its properties and use it, understand Positive-Negative Definite matrices, nonnegative matrices and their properties, and conditions for the irreducibility of nonnegative matrices. Preliminaries, eigenvalues, eigenvectors and similarity, unitary equivalence and normal matrices, Canonical forms, Hermitian and symmetric matrices, norms for vectors and matrices, location and perturbation of eigenvalues, positive definite matrices, nonnegative matrices.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Vector Spaces, Matrices, Determinants, Rank, Nonsingularity, The Usual Inner Product, Partitioned Matrices pp. 1-18
2 The Eigenvalue-Eigenvector Equation, The Characteristic Polynomial, Similarity pp. 33-57
3 Unitary Matrices, Unitary Equivalence pp. 65-78
4 Schur’s Unitary Triangularization Theorem, Normal Matrices pp. 79-111
5 The Jordan Canonical Form, Polynomials and Matrices, The Minimal Polynomial pp. 119-149
6 Triangular Factorization, LU Decomposition pp. 158-166
7 Hermitian Matrices, Properties and Characterizations of Hermitian Matrices, Complex Symmetric Matrices pp. 167-217
8 Defining Properties of Vector Norms and Inner Products, Examles of Vector Norms, Algebraic Properties of Vector Norms pp. 257-268
9 Matrix Norms, Vector Norms on Matrices, Errors in Inverses and Solutions of Linear Systems pp. 290-342
10 Gersgorin Discs, Perturbation Theorems, Other Inclusion Regions pp. 343-390
11 Positive Definite Matrices, Their Properties and Characterizations pp. 391-410
12 The Polar Form and The SVD, The Schur Product Form, Simultaneous Diagonalization pp. 411-468
13 Nonnegative Matrices; Inequalities and Generalities, Positive Matrices pp. 487-502
14 Nonnegative Matrices, Irreducible Nonnnegative Matrices pp. 503-514
15 General Review
16 Final Exam

### Sources

Course Book 1. Matrix Analysis, R.A.Horn & C.R.Johnson, Cambridge University Press, 1991. 2. 1- Matrix Theory; Basic Results and Techniques, By F.Zhang, Springer, 2011 3. 2- Elementary Linear Algebra, B.Kolman &D.R.Hill, 9th edition, Prentice Hall, 2008.

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 5 10
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 55
Final Exam/Final Jury 1 35
Toplam 8 100
 Percentage of Semester Work 65 35 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area X
2 Can apply gained knowledge and problem solving abilities in inter-disciplinary research X
3 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary X
4 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study X
5 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework X
6 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility X
7 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation X
8 To have an oral and written communication ability in at least one of the common foreign languages ("European Language Portfolio Global Scale", Level B2) X
9 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge X
10 Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach. X
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications. X

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class 16 3 48
Presentation/Seminar Prepration
Project
Report
Homework Assignments 5 6 30
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 15 30
Prepration of Final Exams/Final Jury