Robert Gardner's Online Theory of Matrices

Online Theory of Matrices
with online videos Dr. Bob's Online University
(Not an actual university)
Robert 'Dr. Bob' Gardner

This website includes links to class notes and supplements (in PDF) used in the teaching of East Tennessee State University's graduate-only level Theory of Matrices (MATH 5090). Links are also available for video presentations of the notes. The videos have been used as of the teaching of online Theory of Matrices, starting in summer 2020. The material is based closely on James E. Gentle's Matrix Algebra: Theory, Computations, and Applications in Statistics, Springer (2007):

Gentle's Matrix Algebra: Theory, Computations, and Applications in Statistics book, 1st edition

Robert "Dr. Bob" Gardner can be reached by e-mail at: gardnerr@etsu.edu. His office is in Gilbreath Hall, Room 308F on the ETSU campus.

The following notes and supplements are all in PDF. The videos can be streamed from YouTube and should remain permanently available. (The videos previously on ETSU's Panopto host site have been, or may soon be, deleted.)

Chapter 1. Basic Vectors/Matrix Structure and Notation.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
Chapter 1. Basic Vector/Matrix Structure and Notation	Chapter 1 Notes	-	Chapter 1 Video (9:13)

Chapter 2. Vectors and Vector Spaces.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
2.1. Operations on Vectors	2.1 Notes	2.1 Supplement 2.1 Supplement Print File	2.1 Video, Part 1 (53:53) 2.1 Video, Part 2 (58:08) 2.1 Video, Part 3 (1:04:24)
2.2. Cartesian Coordinates and Geometrical Properties of Vectors	2.2 Notes	2.2 Supplement 2.2 Supplement Print File	2.2 Video, Part 1 (36:50) 2.2 Video, Part 2 (54:41)
2.3. Centered Vectors and Variances and Covariances of Vectors	2.3 Notes	2.3 Supplement 2.3 Supplement Print File	2.3 Video (46:54)

Chapter 3. Basic Properties of Matrices.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
3.1. Basic Definitions and Notation	3.1 Notes	3.1 Supplement 3.1 Supplement Print File	3.1 Video, Part 1 (59:16) 3.1 Video, Part 2 (49:52) 3.1 Video, Part 3 (45:30) 3.1 Video, Part 4 (1:00:04) 3.1 Video, Part 5 (58:15)
3.2. Multiplication of Matrices and Multiplication of Vectors and Matrices	3.2 Notes	3.2 Supplement 3.2 Supplement Print File	3.2 Video, Part 1 (46:23) 3.2 Video, Part 2 (28:15) 3.2 Video, Part 3 (1:17:27)
3.3. Matrix Rank and the Inverse of a Full Rank Matrix	3.3 Notes	3.3 Supplement 3.3 Supplement Print File	3.3 Video, Part 1 (59:41) 3.3 Video, Part 2 (25:16) 3.3 Video, Part 3 (59:41) 3.3 Video, Part 4 (47:41) 3.3 Video, Part 5 (21:28)
3.4. More on Partitioned Square Matrices: The Schur Complement	3.4 Notes	3.4 Supplement 3.4 Supplement Print File	3.4 Video (35:45)
3.5. Linear Systems of Equations	3.5 Notes	3.5 Supplement 3.5 Supplement Print File	3.5 Video, Part 1 (51:32) 3.5 Video, Part 2 (27:32)
3.6. Generalized Inverses	3.6 Notes	3.6 Supplement 3.6 Supplement Print File	3.6 Video, Part 1 (23:03) 3.6 Video, Part 2 (19:10)
3.7. Orthogonality	3.7 Notes	3.7 Supplement 3.7 Supplement Print File	3.7 Video (40:19)
3.8. Eigenanalysis; Canonical Factorizations	3.8 Notes	3.8 Supplement 3.8 Supplement Print File	3.8 Video, Part 1 (29:08) 3.8 Video, Part 2 (1:08:34) 3.8 Video, Part 3 (41:40) 3.8 Video, Part 4 (1:28:08) 3.8 Video, Part 5 (1:24:13)
3.9. Matrix Norms	3.9 Notes	3.9 Supplement 3.9 Supplement Print File	3.9 Video, Part 1 (49:39) 3.9 Video, Part 2 (1:12:16)
3.10. Approximation of Matrices	3.10 Notes	-	-

Chapter 4. Vector/Matrix Derivatives and Integrals.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
4.1. Basics of Differentiation	4.1 Notes	-	-
4.2. Types of Differentiation	4.2 Notes	4.2 Supplement 4.2 Supplement Print File	-
4.3. Optimization of Functions (Partial)	4.3 Notes	-	-
4.4. Multiparameter Likelihood Functions	4.4 Notes	-	-
4.5. Integration and Expectation	4.5 Notes	4.5 Supplement 4.5 Supplement Print File	-

Chapter 5. Matrix Transformations and Factorizations.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
5.1. Transformations by Orthogonal Matrices	5.1 Notes	-	5.1 Video (6:07)
5.2. Geometric Transformations	5.2 Notes	-	5.2 Video (44:23)
5.3. Householder Transformations (Reflections)	5.3 Notes	5.3 Supplement 5.3 Supplement Print File	5.3 Video (31:32)
5.4. Givens Transformations (Rotations)	5.4 Notes	-	5.4 Video (27:01)
5.5. Factorizations of Matrices	5.5 Notes	-	5.5 Video (4:12)
5.6. LU and LDU Factorizations	5.6 Notes	5.6 Supplement 5.6 Supplement Print File	5.6 Video (39:06)
5.7. QR Factorization	5.7 Notes	5.7 Supplement 5.7 Supplement Print File	5.7 Video (22:57)
5.8. Singular Value Factorization	5.8 Notes	-	5.8 Video (3:47)
5.9. Factorizations of Nonnegative Definite Matrices	5.9 Notes	5.9 Supplement 5.9 Supplement Print File	5.9 Video (41:56)
5.10. Incomplete Factorizations	5.10 Notes	-	5.10 Video (2:46)

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Last updated: March 16, 2024.