Lecture 7 of 80Matrix Theory
Properties of rank
Prof. Chandra Murthy · IISc Bangalore
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All the lectures
- Course introduction and properties of matrices
- Vector spaces
- Basis, dimension
- Linear transforms
- Fundamental subspaces of a matrix
- Fundamental theorem of linear algebra
- Properties of rank
- Inner product
- Gram-schmidt algorithm
- Orthonormal matrices definition
- Determinant
- Properties of determinants
- Introduction to norms and inner products
- Vector norms and their properties,
- Applications and equivalence of vector norms
- Summary of equivalence of norms
- Dual norms
- Properties and examples of dual norms
- Matrix norms
- Matrix norms: Properties
- Induced norms
- Induced norms and examples
- Spectral radius
- Properties of spectral radius
- Convergent matrices, Banach lemma
- Recap of matrix norms and Levy-Desplanques theorem
- Equivalence of matrix norms and error in inverses of linear systems
- Errors in inverses of matrices
- Errors in solving systems of linear equations
- Introduction to eigenvalues and eigenvectors
- The characteristic polynomial
- Solving characteristic polynomials, eigenvectors properties
- Similarity
- Diagonalization
- Relationship between eigenvalues of BA and AB
- Eigenvector and principle of biorthogonality
- Unitary matrices
- Properties of unitary matrices
- Unitary equivalence
- Schur's triangularization theorem
- Cayley-Hamilton theorem
- Uses of cayley-hamilton theorem and diagonalizability revisited
- Normal matrices: Definition and fundamental properties
- Fundamental properties of normal matrices
- QR decomposition and canonical forms
- Jordan canonical form
- Determining the Jordan form of a matrix
- Properties of the Jordan canonical form (part 1)
- Properties of the Jordan canonical form (part 2)
- Properties of convergent matrices
- Polynomials and matrices
- Other canonical forms and factorization of matrices: Gaussian elimination & LU factorization
- LU decomposition
- LU decomposition with pivoting
- Solving pivoted system and LDM decomposition
- Cholesky decomposition and uses
- Hermitian and symmetric matrix
- Properties of hermitian matrices
- Variational characterization of Eigenvalues: Rayleigh-Ritz theorem
- Variational characterization of eigenvalues (continued)
- Courant-Fischer theorem
- Summary of Rayliegh-Ritz and Courant-Fischer theorems
- Weyl's theorem
- Positive semi-definite matrix, monotonicity theorem and interlacing theorems
- Interlacing theorem I
- Interlacing theorem II (Converse)
- Interlacing theorem (Continued)
- Eigenvalues: Majorization theorem and proof
- Location and perturbation of Eigenvalues Part1: Dominant diagonal theorem
- Location and perturbation of Eigenvalues Part2: Gersgorin's theorem
- Implications of Gersgorin disc theorem, condition of eigenvalues
- Condition of eigenvalues for diagonalizable matrices
- Perturbation of eigenvalues Birkhoff's theorem Hoffman - Weilandt theorem
- Singular value definition and some remarks.
- Proof of singular value decomposition theorem.
- Partitioning the SVD
- Properties of SVD
- Generalized inverse of matrices
- Least squares
- Constrained least squares