E2 212: Matrix Theory: Video Lectures and Notes
Lecture 1: Basic definitions, vector space, basis.
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Lecture 2: Basis, dimension, linear transforms, fundamental subspaces associated with a linear transform.
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Lecture 3: Fundamental subspaces associated with a linear transform, rank.
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Lecture 4: Rank, inner product, Cauchy-Schwarz inequality.
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Lecture 5: Gram-Schmidt algorithm, determinants.
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Lecture 6: Determinants.
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Lecture 7: Properties of determinants, norms.
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Lecture 8: Properties of norms.
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Lecture 9: Dual norms, matrix norms.
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Lecture 10: Properties of matrix norms, induced norms.
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Lecture 11: Induced norms, spectral radius.
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Lecture 12: Properties of spectral radius, Banach lemma.
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Lecture 13: Levy-Desplanques theorem, equivalence of matrix norms, errors in inverses of linear systems.
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Lecture 14: Errors in inverses, errors in solving systems of linear equations.
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Lecture 15: Eigenvalues and eigenvectors, the characteristic polynomial.
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Lecture 16: Similar matrices, diagonalizability.
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Lecture 17: Simultaneous diagonalizability, eigenvectors, principle of biorthogonality.
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Lecture 18: Unitary matrices, unitary equivalence.
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Lecture 19: Schur's unitary triangularization theorem, Cayley-Hamilton theorem.
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Lecture 20: Uses of Cayley-Hamilton theorem, diagonalizability revisited, normal matrices.
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Lecture 21: Properties of normal matrices, QR decomposition, canonical forms.
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Lecture 22: Jordan canonical form.
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Lecture 23: Properties of the Jordan canonical form, convergent matrices.
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Lecture 24: Properties of convergent matrices, polynomials and matrices, Gaussian elimination.
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Lecture 25: Gauss transforms, LU decomposition.
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Lecture 26: LU decomposition with pivoting, LDM, LDLt decomposition.
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Lecture 27: Cholesky decomposition, uses, Hermitian and symmetric matrices.
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Lecture 28: Properties of Hermitian matrices, variational characterization of eigenvalues, Rayleigh-Ritz theorem.
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Lecture 29: Courant-Fischer theorem.
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Lecture 30: Weyl's theorem, eigenvalues of bordered matrices.
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Lecture 31: Interlacing theorems for Hermitian matrices.
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Lecture 32: Further interlacing theorems, majorization of eigenvalues.
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Lecture 33: Location and perturbation of eigenvalues: dominant diagonal theorem, Gersgorin disc theorem.
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Lecture 34: Properties of Gersgorin discs, condition of eigenvalues.
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Lecture 35: Perturbation of eigenvalues, Birkhoff's theorem, Hoffman-Weilandt theorem.
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Lecture 36: Singular value decomposition (SVD).
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Lecture 37: Properties of the SVD, generalized inverses of matrices.
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Lecture 38: Least squares problems, generalized SVD theorem, constrained least squares.
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