Eigenvalues, Eigenvectors and Diagonalization of Matrices - Linear Algebra (Undergraduate Advanced)

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.

26

17 hrs

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.
[OAU, Ife] MTH 202: Mathematical Methods II
[OAU, Ife] 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.

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.

[OAU, Ife] CHE 305: Engineering Analysis I
[OAU, Ife] 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.

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

1. Introduction
2

Welcome and course outline. Review of elementary matrix concepts.

Chapter lessons

1-1. Welcome
9:21

Welcome to the course and outline of course.

1-2. Matrices
39:13

Review of some matrix concepts required for the course.

2. Eigenvalues and Eigenvectors
9
2

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

Chapter lessons

2-1. Polynomials of matrices
14:39

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

2-2. Characteristic polynomials
19:00

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

2-3. Principal minors
19:12

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

2-4. Characteristic polynomials of degrees 2 and 3
19:51

Calculating characteristic polynomials of matrices of orders 2 and 3.

2-5. Characteristic polynomial of degree n
19:56

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

2-6. Cayley-Hamilton theorem
22:27

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

2-7. The eigenvalue problem
26:40

Formal definition of eigenvalues and eigenvectors.

2-8. Multiplicity of eigenvalues
36:35

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

2-9. Properties
13:15

Properties of eigenvalues and eigenvectors of matrices.

3. Diagonalization of Matrices
6
2

Diagonalization of matrices; evaluating polynomials and transcendentals of matrices.

Chapter lessons

3-1. Similar matrices
20:19

Characteristic polynomial and eigenvalues of similar matrices.

3-2. Powers of diagonal matrices
13:10

Evaluating powers of diagonal matrices and matrices similar to them.

3-3. Definition
24:19

Meaning of diagonalization of matrices.

3-4. Diagonalizability
13:36

When are matrices diagonalizable?

3-5. Polynomials of matrices
16:46

Evaluating polynomials of matrices by diagonalization.

3-6. Transcendentals of matrices
22:15

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

4. Symmetric Matrices
3
2

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

Chapter lessons

4-1. Special matrices
33:04

Review of symmetric, orthogonal and orthonormal matrices.

4-2. Diagonalizing symmetric matrices
37:18

Theorems on diagonalization of symmetric matrices.

4-3. Gram-Schmidt procedure
21:14

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

5. Quadratic and Canonical Forms
4
2

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

Chapter lessons

5-1. Definitions
13:07

Meaning of quadratic and canonical forms.

5-2. The coefficient matrix
18:54

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

5-3. Transformation to canonical forms
16:41

Algorithm for transformation of quadratic forms to canonical forms.

5-4. Applications to conics
15:29

Applications of quadratic and canonical forms to conic sections.