Matrix Theory (English)

   Wednesday, May 21, 2025   

Starting a new Lecture Notes Series on Matrix Theory

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Matrix Theory By Lecture Notes together!

Lecture 1: Course introduction and properties of matrices

Lecture 2: Vector spaces

Lecture 3: Basis, dimension

Lecture 4: Linear transforms

Lecture 5: Fundamental subspaces of a matrix

Lecture 6: Fundamental theorem of linear algebra

Lecture 7: Properties of rank

Lecture 8: Inner product

Lecture 9: Gram-schmidt algorithm,

Lecture 10: Orthonormal matrices definition

Lecture 11: Determinant

Lecture 12: Properties of determinants

Lecture 13: Introduction to norms and inner products

Lecture 14: Vector norms and their properties

Lecture 15: Applications and equivalence of vector norms

Lecture 16: Summary of equivalence of norms

Lecture 17: Dual norms

Lecture 18: Properties and examples of dual norms

Lecture 19: Matrix norms

Lecture 20: Matrix norms: Properties

Lecture 21: Induced norms

Lecture 22: Induced norms and examples

Lecture 23: Spectral radius

Lecture 24: Properties of spectral radius

Lecture 25: Convergent matrices, Banach lemma

Lecture 26: Recap of matrix norms and Levy-Desplanques theorem

Lecture 27: Equivalence of matrix norms and error in inverses of linear systems

Lecture 28: Errors in inverses of matrices

Lecture 29: Errors in solving systems of linear equations

Lecture 30: Introduction to eigenvalues and eigenvectors

Lecture 31: The characteristic polynomial

Lecture 32: Solving characteristic polynomials, eigenvectors properties

Lecture 33: Similarity

Lecture 34: Diagonalization

Lecture 35: Relationship between eigenvalues of BA and AB

Lecture 36: Eigenvector and principle of biorthogonality

Lecture 37: Unitary matrices

Lecture 38: Properties of unitary matrices

Lecture 39: Unitary equivalence

Lecture 40: Schur's triangularization theorem

Lecture 41: Cayley-Hamilton theorem

Lecture 42: Uses of cayley-hamilton theorem and diagonalizability revisited

Lecture 43: Normal matrices: Definition and fundamental properties

Lecture 44: Fundamental properties of normal matrices

Lecture 45: QR decomposition and canonical forms

Lecture 46: Jordan canonical form

Lecture 47: Determining the Jordan form of a matrix

Lecture 48: Properties of the Jordan canonical form (part 1)

Lecture 49: Properties of the Jordan canonical form (part 2)

Lecture 50: Properties of convergent matrices

Lecture 51: Polynomials and matrices

Lecture 52: Other canonical forms and factorization of matrices: Gaussian elimination & LU factorization

Lecture 53: LU decomposition

Lecture 54: LU decomposition with pivoting

Lecture 55: Solving pivoted system and LDM decomposition

Lecture 56: Cholesky decomposition and uses

Lecture 57: Hermitian and symmetric matrix

Lecture 58: Properties of hermitian matrices

Lecture 59: Variational characterization of Eigenvalues: Rayleigh-Ritz theorem

Lecture 60: Variational characterization of eigenvalues (continued)

Lecture 61: Courant-Fischer theorem

Lecture 62: Summary of Rayliegh-Ritz and Courant-Fischer theorems

Lecture 63: Weyl's theorem

Lecture 64: Positive semi-definite matrix, monotonicity theorem and interlacing theorems

Lecture 65: Interlacing theorem I

Lecture 66: Interlacing theorem II (Converse)

Lecture 67: Interlacing theorem (Continued)

Lecture 68: Eigenvalues: Majorization theorem and proof

Lecture 69: Location and perturbation of Eigenvalues Part1: Dominant diagonal theorem

Lecture 70: Location and perturbation of Eigenvalues Part2: Gersgorin's theorem

Lecture 71: Implications of Gersgorin disc theorem, condition of eigenvalues

Lecture 72: Condition of eigenvalues for diagonalizable matrices

Lecture 73: Perturbation of eigenvalues Birkhoff's theorem Hoffman - Weilandt theorem

Lecture 74: Singular value definition and some remarks.

Lecture 75: Proof of singular value decomposition theorem.

Lecture 76: Partitioning the SVD

Lecture 77: Properties of SVD

Lecture 78: Generalized inverse of matrices

Lecture 79: Least squares

Lecture 80: Constrained least squares