E2 212: Matrix Theory: Video Lectures and Notes

Lecture 1: Basic definitions, vector space, basis. Video, Notes.

Lecture 2: Basis, dimension, linear transforms, fundamental subspaces associated with a linear transform. Video, Notes.

Lecture 3: Fundamental subspaces associated with a linear transform, rank. Video, Notes.

Lecture 4: Rank, inner product, Cauchy-Schwarz inequality. Video, Notes.

Lecture 5: Gram-Schmidt algorithm, determinants. Video, Notes.

Lecture 6: Determinants. Video, Notes.

Lecture 7: Properties of determinants, norms. Video, Notes.

Lecture 8: Properties of norms. Video, Notes.

Lecture 9: Dual norms, matrix norms. Video, Notes.

Lecture 10: Properties of matrix norms, induced norms. Video, Notes.

Lecture 11: Induced norms, spectral radius. Video, Notes.

Lecture 12: Properties of spectral radius, Banach lemma. Video, Notes.

Lecture 13: Levy-Desplanques theorem, equivalence of matrix norms, errors in inverses of linear systems. Video, Notes.

Lecture 14: Errors in inverses, errors in solving systems of linear equations. Video, Notes.

Lecture 15: Eigenvalues and eigenvectors, the characteristic polynomial. Video, Notes.

Lecture 16: Similar matrices, diagonalizability. Video, Notes.

Lecture 17: Simultaneous diagonalizability, eigenvectors, principle of biorthogonality. Video, Notes.

Lecture 18: Unitary matrices, unitary equivalence. Video, Notes.

Lecture 19: Schur’s unitary triangularization theorem, Cayley-Hamilton theorem. Video, Notes.

Lecture 20: Uses of Cayley-Hamilton theorem, diagonalizability revisited, normal matrices. Video, Notes.

Lecture 21: Properties of normal matrices, QR decomposition, canonical forms. Video, Notes.

Lecture 22: Jordan canonical form. Video, Notes.

Lecture 23: Properties of the Jordan canonical form, convergent matrices. Video, Notes.

Lecture 24: Properties of convergent matrices, polynomials and matrices, Gaussian elimination. Video, Notes.

Lecture 25: Gauss transforms, LU decomposition. Video, Notes.

Lecture 26: LU decomposition with pivoting, LDM, LDLt decomposition. Video, Notes.

Lecture 27: Cholesky decomposition, uses, Hermitian and symmetric matrices. Video, Notes.

Lecture 28: Properties of Hermitian matrices, variational characterization of eigenvalues, Rayleigh-Ritz theorem. Video, Notes.

Lecture 29: Courant-Fischer theorem. Video, Notes.

Lecture 30: Weyl’s theorem, eigenvalues of bordered matrices. Video, Notes.

Lecture 31: Interlacing theorems for Hermitian matrices. Video, Notes.

Lecture 32: Further interlacing theorems, majorization of eigenvalues. Video, Notes.

Lecture 33: Location and perturbation of eigenvalues: dominant diagonal theorem, Gersgorin disc theorem. Video, Notes.

Lecture 34: Properties of Gersgorin discs, condition of eigenvalues. Video, Notes.

Lecture 35: Perturbation of eigenvalues, Birkhoff’s theorem, Hoffman-Weilandt theorem. Video1,Video2, Notes.

Lecture 36: Singular value decomposition (SVD). Video, Notes.

Lecture 37: Properties of the SVD, generalized inverses of matrices. Video, Notes.

Lecture 38: Least squares problems, generalized SVD theorem, constrained least squares. Video, Notes.