Eigenvalues, Eigenvectors and Diagonalization of Matrices

This course provides a clear and structured exploration of eigenvalues, eigenvectors, and diagonalization, focusing on both theory and real-world applications. You’ll learn how to compute eigenvalues and eigenvectors, understand their geometric significance, and apply diagonalization to simplify complex matrix operations. Topics include linear transformations, dynamical systems, and applications in physics, engineering, and machine learning. The course is designed for students, engineers, and data scientists seeking a strong foundation in matrix methods. By the end, you'll confidently apply these concepts in problem-solving and computational modeling.

17 hrs

$ 10.00

Payment required for enrolment

Enrolment valid for 12 months

This course is also part of the following learning tracks. You may join a track to gain comprehensive knowledge across related courses.

MTH 202: Mathematical Methods II

Comprehensive treatise of advanced mathematics covering vector calculus, complex numbers, linear vector spaces, linear maps, matrices, eigenvalues and eigenvectors. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

CHE 305: Engineering Analysis I

Advanced engineering mathematics covering solid analytical geometry, integrals, scalar and vector fields, matrices and determinants and complex variables. Curated for third-year students of engineering at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Course Chapters

Introduction

Welcome and course outline. Review of elementary matrix concepts.

Chapter lessons

1.Welcome9:21

Welcome to the course and outline of course.

2.Matrices39:13

Review of some matrix concepts required for the course.

Eigenvalues and Eigenvectors

Meaning, operations and properties of eigenvalues and eigenvectors of matrices.

Chapter lessons

1.Polynomials of matrices14:39

Meaning of polynomials of matrices, and how to evaluate them by direct multiplication.

2.Characteristic polynomials19:00

Meaning of characteristic polynomials of matrices and how to calculate them.

3.Principal minors19:12

Meaning of principal minors in general, and principal minors of a given order.

4.Characteristic polynomials of degrees 2 and 319:51

Calculating characteristic polynomials of matrices of orders 2 and 3.

5.Characteristic polynomial of degree n19:56

General expression for the characteristic polynomial of a given square matrix of order n, in terms of its principal minors.

6.Cayley-Hamilton theorem22:27

Statement of Cayley-Hamilton theorem; calculating the inverse of a matrix using Cayley-Hamilton theorem.

7.The eigenvalue problem26:40

Formal definition of eigenvalues and eigenvectors.

8.Multiplicity of eigenvalues36:35

Meaning of algebraic and geometric multiplicities of eigenvalues - with worked examples.

9.Properties13:15

Properties of eigenvalues and eigenvectors of matrices.

10.Worked examples (1)1:04:51

Worked examples on calculating eigenvalues and eigenvectors.

11.Worked examples (2)23:55

Worked examples on calculating eigenvalues and eigenvectors.

Diagonalization of Matrices

Diagonalization of matrices; evaluating polynomials and transcendentals of matrices.

Chapter lessons

1.Similar matrices20:19

Characteristic polynomial and eigenvalues of similar matrices.

2.Powers of diagonal matrices13:10

Evaluating powers of diagonal matrices and matrices similar to them.

3.Definition24:19

Meaning of diagonalization of matrices.

4.Diagonalizability13:36

When are matrices diagonalizable?

5.Worked examples (1)31:12

Worked examples on diagonalization of matrices.

6.Polynomials of matrices16:46

Evaluating polynomials of matrices by diagonalization.

7.Transcendentals of matrices22:15

Evaluating exponentials, logarithms, sines, cosines, etc., of matrices - by diagonalization.

8.Worked examples (2)29:41

More worked examples on polynomials and transcendentals of matrices.

Symmetric Matrices

Diagonalization of symmetric matrices - orthogonal diagonalizing matrix, the Gram-Schmidt orthogonalization procedure, and related concepts.

Chapter lessons

1.Special matrices33:04

Review of symmetric, orthogonal and orthonormal matrices.

2.Diagonalizing symmetric matrices37:18

Theorems on diagonalization of symmetric matrices.

3.Gram-Schmidt procedure21:14

Gram-Schmidt procedure for obtaining an orthonormal set of vectors from a linearly-independent set of vectors.

4.Worked examples (1)26:44

Worked examples on Gram-Schmidt orthonormalization algorithm.

5.Worked examples (2)53:59

Worked examples on diagonalization of symmetric matrices involving the use of Gram-Schmidt procedure.

Quadratic and Canonical Forms

Quadratic and canonical forms; transformations using symmetric matrices and orthogonal diagonalizing matrices.

Chapter lessons

1.Definitions13:07

Meaning of quadratic and canonical forms.

2.The coefficient matrix18:54

How to obtain the [symmetric] coefficient matrix for quadratic and canonical forms.

3.Transformation to canonical forms16:41

Algorithm for transformation of quadratic forms to canonical forms.

4.Worked examples (1)39:55

Worked examples on transformation of quadratic form to canonical forms.

5.Applications to conics15:29

Applications of quadratic and canonical forms to conic sections.

6.Worked examples (2)40:06

Worked examples on applications of quadratic and canonical forms to conic sections.