Eigenvalues, Eigenvectors and Diagonalization of Matrices - Linear Algebra (Undergraduate Advanced)
26
17 hrs
MTH 202: Mathematical Methods IIComprehensive 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.
CHE 305: Engineering Analysis IAdvanced 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.
MTH 205: Linear Algebra IIAdvanced linear algebra is the mathematical backbone of modern data science, engineering, and physics. This learning track delivers the rigorous MTH 205 curriculum based on NUC CCMAS standards, focusing on sophisticated matrix analysis and practical computational methods critical for solving complex technical problems.
This programme is targeted at undergraduates in mathematics, engineering, and computer science requiring a deep command of advanced matrix theory. It is equally essential for data scientists and engineers seeking a rigorous theoretical foundation for machine learning algorithms, cryptography, and complex system modelling.
You will master matrix manipulations to solve linear systems and compute determinants and inverses efficiently using various methods including software like Python and MATLAB. You will gain competence in determining eigenvalues and eigenvectors, applying diagonalization to analyze the stability of dynamical systems, and working with quadratic and canonical forms. Completion establishes the critical mathematical expertise required for advanced studies in multivariate statistics, differential equations, and algorithmic development.
Advanced linear algebra is the mathematical backbone of modern data science, engineering, and physics. This learning track delivers the rigorous MTH 205 curriculum based on NUC CCMAS standards, focusing on sophisticated matrix analysis and practical computational methods critical for solving complex technical problems. This programme is targeted at undergraduates in mathematics, engineering, and computer science requiring a deep command of advanced matrix theory. It is equally essential for data scientists and engineers seeking a rigorous theoretical foundation for machine learning algorithms, cryptography, and complex system modelling. You will master matrix manipulations to solve linear systems and compute determinants and inverses efficiently using various methods including software like Python and MATLAB. You will gain competence in determining eigenvalues and eigenvectors, applying diagonalization to analyze the stability of dynamical systems, and working with quadratic and canonical forms. Completion establishes the critical mathematical expertise required for advanced studies in multivariate statistics, differential equations, and algorithmic development.
Course Chapters
1. Introduction2
2. Eigenvalues and Eigenvectors92
Meaning, operations and properties of eigenvalues and eigenvectors of matrices.
Chapter lessons
2-1. Polynomials of matrices14:39
2-2. Characteristic polynomials19:00
2-3. Principal minors19:12
2-4. Characteristic polynomials of degrees 2 and 319:51
Calculating characteristic polynomials of matrices of orders 2 and 3.
2-5. 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.
2-6. Cayley-Hamilton theorem22:27
Statement of Cayley-Hamilton theorem; calculating the inverse of a matrix using Cayley-Hamilton theorem.
2-7. The eigenvalue problem26:40
Formal definition of eigenvalues and eigenvectors.
2-8. Multiplicity of eigenvalues36:35
Meaning of algebraic and geometric multiplicities of eigenvalues - with worked examples.
2-9. Properties13:15
Properties of eigenvalues and eigenvectors of matrices.
3. Diagonalization of Matrices62
Diagonalization of matrices; evaluating polynomials and transcendentals of matrices.
Chapter lessons
3-1. Similar matrices20:19
Characteristic polynomial and eigenvalues of similar matrices.
3-2. Powers of diagonal matrices13:10
Evaluating powers of diagonal matrices and matrices similar to them.
3-3. Definition24:19
Meaning of diagonalization of matrices.
3-4. Diagonalizability13:36
When are matrices diagonalizable?
3-5. Polynomials of matrices16:46
Evaluating polynomials of matrices by diagonalization.
3-6. Transcendentals of matrices22:15
Evaluating exponentials, logarithms, sines, cosines, etc., of matrices - by diagonalization.
4. Symmetric Matrices32
Diagonalization of symmetric matrices - orthogonal diagonalizing matrix, the Gram-Schmidt orthogonalization procedure, and related concepts.
Chapter lessons
4-1. Special matrices33:04
Review of symmetric, orthogonal and orthonormal matrices.
4-2. Diagonalizing symmetric matrices37:18
Theorems on diagonalization of symmetric matrices.
4-3. Gram-Schmidt procedure21:14
Gram-Schmidt procedure for obtaining an orthonormal set of vectors from a linearly-independent set of vectors.
5. Quadratic and Canonical Forms42
Quadratic and canonical forms; transformations using symmetric matrices and orthogonal diagonalizing matrices.
Chapter lessons
5-1. Definitions13:07
Meaning of quadratic and canonical forms.
5-2. The coefficient matrix18:54
How to obtain the [symmetric] coefficient matrix for quadratic and canonical forms.
5-3. Transformation to canonical forms16:41
Algorithm for transformation of quadratic forms to canonical forms.
5-4. Applications to conics15:29
Applications of quadratic and canonical forms to conic sections.