Linear algebra

la linear algebra
nla numerical linear algebra

Linear algebra

Main reference

• Lectures from Prof. Stephen Boyd (Stanford)

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• Matrix-vector multiplication interpreted as transformation and translation
• Change of basis based on two interpretations

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• Matrix-vector multiplication as the only way of expressing linear function
• Affine function

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• Algebraic definition of dot (inner) product
• Dot product as projection
• First look of projection matrix

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• Geometric intuition of determinant
• Determinant and cross product

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• Interpretations of matrix-vector multiplication
• Interpretations of matrix-matrix multiplication

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• Vector space and subspace
• Independent set of vectors
• Basis of a vector space

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• Nullspace of a matrix
• Zero nullspace and equivalent statement: left inverse, independent columns, determinant,...

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• Column space of a matrix
• Onto matrix and equivalent statement: right inverse, independent rows, determinant,...
• Left inverse and right inverse

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• Inverse of square matrices
• Left and right invertibility
• Left and right inverse are unique and equal
• Dual basis

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• Rank of a matrix
• Column rank equals row rank
• Full rank matrices

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• Euclidean norm of a vector
• p-norm of a vector
• Cauchy-Schwarz inequality

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• Orthonormal basis
• Equivalence between transpose and inverse
• Preservation of norm, inner product and angle
• Orthogonal matrix
• Projection of a vector onto subspace
• Two interpretations of projection

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• Find orthonormal basis for an independent set of vectors
• Basic Gram-Schmidt procedure based on sequential orthogonalization

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• Find orthonormal basis for a set of vectors (independent or not)
• General Gram-Schmidt procedure and QR factorization

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• Full QR factorization
• Relationship among four subspaces of matrices through QR
• Bessel's inequality

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• Derivative, Jacobian, and gradient
• Chain rule for 1st derivative
• Hessian
• Chain rule for 2nd derivative

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• Least squares and left inverse
• Geometric interpretation and projection matrix
• Least squares through QR factorization
• Properties of projection matrices

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• Multi-objective least squares and regularization
• Tikhonov regularization

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• Least squares problem with equality constraints
• Least norm problem
• Right inverse as solution to least norm problem
• Derivation of least norm solution through method of Lagrange multipliers
• Intuition behind method of Lagrange multipliers

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• Optimality conditions for equality-constrained least squares
• KKT equations and invertibility of KKT matrix

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• Verification of solution obtained from KKT equations
• The solution is also unique

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• Definition of matrix exponential

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• Left and right eigenvectors
• Real matrices have conjugate symmetry for eigenvalues and eigenvectors
• Characteristic polynomial
• Basic properties of eigenvalues
• Markov chain example

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• Matrices with independent set of eigenvectors are diagonalizable
• Not all square matrices are diagonalizable
• Matrices with distinct eigenvalues are diagonalizable
• The other way is not true
• Diagonalization simplifies expressions: resolvent, powers, exponential,...
• Diagonalization simplifies linear relation
• Diagonalization and left eigenvectors
• Left and right eigenvectors as dual basis

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• Jordan canonical form generalizes diagonalization of square matrices
• Determinant of a matrix is product of all its eigenvalues
• Generalized eigenvectors
• Cayley-Hamilton theorem
• Corollary and intuition

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• Symmetric matrices have real eigenvalues
• Symmetirc matrices have orthogonal eigenvectors

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• Similarity transformation
• Things preserved under similarity transformation (characteristic polynomial, eigenvalues, determinant, matrix rank...)

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• Quadratic form
• Uniqueness of quadratic form
• Upper and lower bound of quadratic form
• Positive semidefinite and positive definite matrices
• Simple inequality related to largest/smallest eigenvalue

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• Gain of a matrix
• Matrix norm as the largest gain
• Nullspace and zero gain

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• Singular value decomposition (SVD) as a more complete picture of gain of matrix
• Singular vectors
• Input direction and sensitivity
• Output direction
• Comparison to eigendecomposition

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• Left and right inverses computed via SVD

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• Full SVD
• Full SVD simplifies linear relation

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• Error estimation in linear model via SVD
• Two types of ellipsoids

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• Sensitivity of linear equations to data error
• Condition number
• Condition number of one
• Tightness of condition number bound

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• SVD as optimal low-rank approximation
• Singular value as distance to singularity
• Model simplification with SVD

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• Example: position estimation from ranges
• Nonlinear least squares (NLLS)
• Gauss-Newton algorithm

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• Example: position estimation from ranges
• Issues with Gauss-Newton algorithm
• Levenberg-Marquardt algorithm for NLLS

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• Example: k-nearest neighbours classifier

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• Example: principal component analysis (PCA)
• Recap SVD
• Covariance in data
• Principal components

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• Example: k-means clustering

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• Example: tomography with regularized least squares
• Coordinates of image pixels
• Length of intersection between ray and pixel
• Regularized least squares for underdetermined system
• Tikhonov regularization vs Laplacian regularization

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• Example: linear quadratic state estimation with constrained least squares

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• Example: polynomial data fitting with constrained least squares

Numerical linear algebra

Main reference

• Numerical Linear Algebra (Lloyd Trefethen and David Bau)

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• Classic Gram-Schmidt (CGS) in rank-one projection form
• Modified Gram-Schmidt (MGS) for numerical stability

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• Householder reflector for QR factorization
• Construction of reflection matrix
• Orthogonality of reflection matrix
• Householder finds full QR factorization

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• Givens rotation for QR factorization
• Construction of rotation matrix
• Orthogonality of rotation matrix
• Givens rotation finds full QR factorization

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• Gaussian elimination for solving linear system of equations
• LU factorization with partial pivoting
• Concept of permutation matrices

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• Cholesky factorization for positive definite matrices
• Use Cholesky to detect non-positive definite matrices
• LDLT factorization for nonsingular symmetric matrices

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• Compute determinant using factorization (Cholesky and LU with partial pivoting)

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• Forward substitution
• Back substitution

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• Condition of a problem
• Absolute and relative condition number
• Condition of matrix-vector multiplication
• Forward and backward error
• Condition number of a matrix
• Condition of a system of equations

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• Double precision system
• Machine precision
• Precision in relative terms
• Cancellation error

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• Conditioning parameters and condition numbers for least squares
• Numerical stability of different QR factorization in solving ill-conditioned least squares

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• Solve linear system of equations with symmetric matrices using Cholesky or LDLT
• Solve generic linear system of equations using LU factorization with partial pivoting
• Block elimination for equations with structured sub-blocks

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• Power iterations to compute dominant eigenvalue for diagonalizable matrices
• Convergence to eigenvector
• Rayleigh quotient and convergence to eigenvalue
• Compute all eigenvalues for symmetric matrices

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• Power iterations for nonsymmetric matrices
• Update of matrix with both left and right eigenvectors
• Update of biorthogonality between left and right eigenvectors
• Compute all eigenvalues for nonsymmetric matrices

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• Schur decomposition
• Obtain Schur form of general matrices using orthogonal iterations
• Upper triangular matrix in Schur form contains eigenvalues in its diagonal
• Computation of eigenvectors based on Schur form

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• QR algorithm as refomulation of orthogonal iterations for general matrices

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• Two-phase approach to compute SVD of a matrix
• Phase one: turn matrix into bidiagonal form
• Phase two: compute SVD of bidiagonal matrix

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• Fixed point method for iterative solution to linear system of equations
• Jacobi method
• Gauss-Seidel method and successive over relaxation (SOR)
• Convergence requirement

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• Characteristic polynomial and minimal polynomial
• Intuition via generalized eigenvectors
• Krylov subspace

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• Hessenberg form of matrices
• Arnoldi iteration for Hessenberg decomposition
• Arnoldi iteration constructs orthonormal basis for successive Krylov subspaces
• Reduced Hessenberg form
• Eigenvalue approximation

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• Lanczos iteration as special case of Arnoldi iteration for symmetric matrices
• Hessenberg form in symmetric case is tridiagonal
• Reorthogonalization for Lanczos iteration

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• Generalized minimal residuals (GMRES) for iterative solution to linear system of equations
• Relation to Arnoldi iteration
• Convergence of GMRES

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• Gradient method for iterative solution to linear system of equations
• Gradient descent and line search for optimal step size
• Conjugate gradient method for positive definite matrices