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Introduction to
Applied Linear Algebra
Vectors, Matrices, and Least Squares
Stephen Boyd
Department of Electrical Engineering
Stanford University
Lieven Vandenberghe
Department of Electrical and Computer Engineering
University of California, Los Angeles
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Daniel and Margriet
Contents
Preface
xiI Vectors
11 Vectors
31.1 Vectors
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Vector addition
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Scalar-vector multiplication
. . . . . . . . . . . . . . . . . . . . . . . . 151.4 Inner product
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.5 Complexity of vector computations
. . . . . . . . . . . . . . . . . . . . 22Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Linear functions
292.1 Linear functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Taylor approximation
. . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3 Regression model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Norm and distance
453.1 Norm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Distance
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3 Standard deviation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.4 Angle
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.5 Complexity
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 Clustering
694.1 Clustering
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 A clustering objective
. . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3 Thek-means algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 Examples
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.5 Applications
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 viiiContents5 Linear independence895.1 Linear dependence
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 Basis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.3 Orthonormal vectors
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.4 Gram{Schmidt algorithm
. . . . . . . . . . . . . . . . . . . . . . . . . 97Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103II Matrices
1056 Matrices
1076.1 Matrices
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2 Zero and identity matrices
. . . . . . . . . . . . . . . . . . . . . . . . 1136.3 Transpose, addition, and norm
. . . . . . . . . . . . . . . . . . . . . . 1156.4 Matrix-vector multiplication
. . . . . . . . . . . . . . . . . . . . . . . . 1186.5 Complexity
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247 Matrix examples
1297.1 Geometric transformations
. . . . . . . . . . . . . . . . . . . . . . . . 1297.2 Selectors
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.3 Incidence matrix
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.4 Convolution
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1448 Linear equations
1478.1 Linear and ane functions
. . . . . . . . . . . . . . . . . . . . . . . . 1478.2 Linear function models
. . . . . . . . . . . . . . . . . . . . . . . . . . 1508.3 Systems of linear equations
. . . . . . . . . . . . . . . . . . . . . . . . 152Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1599 Linear dynamical systems
1639.1 Linear dynamical systems
. . . . . . . . . . . . . . . . . . . . . . . . . 1639.2 Population dynamics
. . . . . . . . . . . . . . . . . . . . . . . . . . . 1649.3 Epidemic dynamics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1689.4 Motion of a mass
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1699.5 Supply chain dynamics
. . . . . . . . . . . . . . . . . . . . . . . . . . 171Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17410 Matrix multiplication
17710.1 Matrix-matrix multiplication
. . . . . . . . . . . . . . . . . . . . . . . 17710.2 Composition of linear functions
. . . . . . . . . . . . . . . . . . . . . . 18310.3 Matrix power
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18610.4 QR factorization
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191Contentsix11 Matrix inverses199
11.1 Left and right inverses
. . . . . . . . . . . . . . . . . . . . . . . . . . . 19911.2 Inverse
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20211.3 Solving linear equations
. . . . . . . . . . . . . . . . . . . . . . . . . . 20711.4 Examples
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21011.5 Pseudo-inverse
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217III Least squares
22312 Least squares
22512.1 Least squares problem
. . . . . . . . . . . . . . . . . . . . . . . . . . . 22512.2 Solution
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22712.3 Solving least squares problems
. . . . . . . . . . . . . . . . . . . . . . 23112.4 Examples
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23913 Least squares data tting
24513.1 Least squares data tting
. . . . . . . . . . . . . . . . . . . . . . . . . 24513.2 Validation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26013.3 Feature engineering
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27914 Least squares classication
28514.1 Classication
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28514.2 Least squares classier
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28814.3 Multi-class classiers
. . . . . . . . . . . . . . . . . . . . . . . . . . . 297Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30515 Multi-objective least squares
30915.1 Multi-objective least squares
. . . . . . . . . . . . . . . . . . . . . . . 30915.2 Control
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31415.3 Estimation and inversion
. . . . . . . . . . . . . . . . . . . . . . . . . 31615.4 Regularized data tting
. . . . . . . . . . . . . . . . . . . . . . . . . . 32515.5 Complexity
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33416 Constrained least squares
33916.1 Constrained least squares problem
. . . . . . . . . . . . . . . . . . . . 33916.2 Solution
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34416.3 Solving constrained least squares problems
. . . . . . . . . . . . . . . . 347Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 xContents17 Constrained least squares applications35717.1 Portfolio optimization
. . . . . . . . . . . . . . . . . . . . . . . . . . . 35717.2 Linear quadratic control
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