Linear Algebra in Action

Contents......Page f005.djvu
Preface......Page f015.djvu
1.1 Preview......Page p001.djvu
1.2 The abstract definition of a vector space......Page p002.djvu
1.3 Some definitions......Page p005.djvu
1.4 Mappings......Page p011.djvu
1.5 Triangular matrices......Page p013.djvu
1.6 Block triangular matrices......Page p016.djvu
1.7 Schur complements......Page p017.djvu
1.8 Other matrix products......Page p019.djvu
2 Gaussian elimination......Page p021.djvu
2.1 Some preliminary observations......Page p022.djvu
2.2 Examples......Page p024.djvu
2.3 Upper echelon matrices......Page p030.djvu
2.4 The conservation of dimension......Page p036.djvu
2.6 Conservation of dimension for matrices......Page p038.djvu
2.7 From U to A......Page p040.djvu
2.8 Square matrices......Page p041.djvu
3.1 Gaussian elimination redux......Page p045.djvu
3.2 Properties of BA and AC......Page p048.djvu
3.3 Extracting a basis......Page p050.djvu
3.4 Computing the coefficients in a basis......Page p051.djvu
3.5 The Gauss-Seidel method......Page p052.djvu
3.6 Block Gaussian elimination......Page p055.djvu
3.7 {0, 1, infty}......Page p056.djvu
3.8 Review......Page p057.djvu
4 Eigenvalues and eigenvectors......Page p061.djvu
4.1 Change of basis and similarity......Page p062.djvu
4.3 Existence of eigenvalues......Page p064.djvu
4.4 Eigenvalues for matrices......Page p066.djvu
4.5 Direct sums......Page p069.djvu
4.6 Diagonalizable matrices......Page p071.djvu
4.7 An algorithm for diagonalizing matrices......Page p073.djvu
4.8 Computing eigenvalues at this point......Page p074.djvu
4.9 Not all matrices are diagonalizable......Page p076.djvu
4.10 The Jordan decomposition theorem......Page p078.djvu
4.11 An instructive example......Page p079.djvu
4.13 More direct sum decompositions......Page p082.djvu
4.14 Verification of Theorem 4.12......Page p084.djvu
4.15 Bibliographical notes......Page p087.djvu
5.1 Functionals......Page p089.djvu
5.2 Determinants......Page p090.djvu
5.3 Useful rules for calculating determinants......Page p093.djvu
5.4 Eigenvalues......Page p097.djvu
5.5 Exploiting block structure......Page p099.djvu
5.6 The Binet-Cauchy formula......Page p102.djvu
5.7 Minors......Page p104.djvu
5.9 Companion matrices......Page p108.djvu
5.10 Circulants and Vandermonde matrices......Page p109.djvu
6 Calculating Jordan forms......Page p111.djvu
6.2 Structure of the nullspaces N_{B^j}......Page p112.djvu
6.3 Chains and cells......Page p115.djvu
6.4 Computing J......Page p116.djvu
6.5 An algorithm for U......Page p117.djvu
6.6 An example......Page p120.djvu
6.7 Another example......Page p122.djvu
6.8 Jordan decompositions for real matrices......Page p126.djvu
6.9 Companion and generalized Vandermonde matrices......Page p128.djvu
7.1 Four inequalities......Page p133.djvu
7.2 Normed linear spaces......Page p138.djvu
7.3 Equivalence of norms......Page p140.djvu
7.4 Norms of linear transformations......Page p142.djvu
7.5 Multiplicative norms......Page p143.djvu
7.6 Evaluating some operator norms......Page p145.djvu
7.7 Small perturbations......Page p147.djvu
7.8 Another estimate......Page p149.djvu
7.9 Bounded linear functionals......Page p150.djvu
7.10 Extensions of bounded linear functionals......Page p152.djvu
7.11 Banach spaces......Page p155.djvu
8.1 Inner product spaces......Page p157.djvu
8.2 A characterization of inner product spaces......Page p160.djvu
8.3 Orthogonality......Page p161.djvu
8.5 Adjoints......Page p163.djvu
8.6 The Riesz representation theorem......Page p166.djvu
8.7 Normal, selfadjoint and unitary transformations......Page p168.djvu
8.8 Projections and direct sum decompositions......Page p170.djvu
8.9 Orthogonal projections......Page p172.djvu
8.10 Orthogonal expansions......Page p174.djvu
8.11 The Gram-Schmidt method......Page p177.djvu
8.12 Toeplitz and Hankel matrices......Page p178.djvu
8.13 Gaussian quadrature......Page p180.djvu
8.14 Bibliographical notes......Page p183.djvu
9 Symmetric, Hermitian and normal matrices......Page p185.djvu
9.1 Hermitian matrices are diagonalizable......Page p186.djvu
9.2 Commuting Hermitian matrices......Page p188.djvu
9.3 Real Hermitian matrices......Page p190.djvu
9.4 Projections and direct sums in F^n......Page p191.djvu
9.6 Normal matrices......Page p195.djvu
9.7 Schur's theorem......Page p198.djvu
9.8 QR factorization......Page p201.djvu
9.9 Areas, volumes and determinants......Page p202.djvu
9.10 Bibliographical notes......Page p206.djvu
10.1 Singular value decompositions......Page p207.djvu
10.2 Complex symmetric matrices......Page p212.djvu
10.3 Approximate solutions of linear equations......Page p213.djvu
10.4 The Courant-Fischer theorem......Page p215.djvu
10.5 Inequalities for singular values......Page p218.djvu
10.6 Bibliographical notes......Page p225.djvu
11.1 Pseudoinverses......Page p227.djvu
11.2 The Moore-Penrose inverse......Page p234.djvu
11.3 Best approximation in terms of Moore-Penrose inverses......Page p237.djvu
12 Triangular factorization and positive definite matrices......Page p239.djvu
12.1 A detour on triangular factorization......Page p240.djvu
12.2 Definite and semidefinite matrices......Page p242.djvu
12.3 Characterizations of positive definite matrices......Page p244.djvu
12.4 An application of factorization......Page p247.djvu
12.5 Positive definite Toeplitz matrices......Page p248.djvu
12.6 Detour on block Toeplitz matrices......Page p254.djvu
12.7 A maximum entropy matrix completion problem......Page p258.djvu
12.8 Schur complements for semidefinite matrices......Page p262.djvu
12.9 Square roots......Page p265.djvu
12.10 Polar forms......Page p267.djvu
12.11 Matrix inequalities......Page p268.djvu
12.12 A minimal norm completion problem......Page p271.djvu
12.13 A description of all solutions to the minimal norm completion problem......Page p273.djvu
12.14 Bibliographical notes......Page p274.djvu
13 Difference equations and differential equations......Page p275.djvu
13.1 Systems of difference equations......Page p276.djvu
13.2 The exponential e^{tA}......Page p277.djvu
13.3 Systems of differential equations......Page p279.djvu
13.4 Uniqueness......Page p281.djvu
13.5 Isometric and isospectral flows......Page p282.djvu
13.6 Second-order differential systems......Page p283.djvu
13.7 Stability......Page p284.djvu
13.9 Strategy for equations......Page p285.djvu
13.10 Second-order difference equations......Page p286.djvu
13.11 Higher order difference equations......Page p289.djvu
13.12 Ordinary differential equations......Page p290.djvu
13.13 Wronskians......Page p293.djvu
13.14 Variation of parameters......Page p295.djvu
14 Vector valued functions......Page p297.djvu
14.1 Mean value theorems......Page p298.djvu
14.2 Taylor's formula with remainder......Page p299.djvu
14.3 Application of Taylor's formula with remainder......Page p300.djvu
14.5 Mean value theorems for vector valued functions of several variables......Page p301.djvu
14.6 Newton's method......Page p304.djvu
14.7 A contractive fixed point theorem......Page p306.djvu
14.8 A refined contractive fixed point theorem......Page p308.djvu
14.9 Spectral radius......Page p309.djvu
14.10 The Brouwer fixed point theorem......Page p313.djvu
14.11 Bibliographical notes......Page p316.djvu
15.1 Preliminary discussion......Page p317.djvu
15.2 The main theorem......Page p319.djvu
15.3 A generalization of the implicit function theorem......Page p324.djvu
15.4 Continuous dependence of solutions......Page p326.djvu
15.5 The inverse function theorem......Page p327.djvu
15.7 An instructive example......Page p329.djvu
15.8 A more sophisticated approach......Page p331.djvu
15.9 Dynamical systems......Page p333.djvu
15.10 Lyapunov functions......Page p335.djvu
15.11 Bibliographical notes......Page p336.djvu
16.1 Classical extremal problems......Page p337.djvu
16.2 Extremal problems with constraints......Page p341.djvu
16.3 Examples......Page p344.djvu
16.5 The conjugate gradient method......Page p349.djvu
16.6 Dual extremal problems......Page p354.djvu
16.7 Bibliographical notes......Page p356.djvu
17.1 Differentiation......Page p357.djvu
17.2 Contour integration......Page p361.djvu
17.3 Evaluating integrals by contour integration......Page p365.djvu
17.4 A short detour on Fourier analysis......Page p370.djvu
17.5 Contour integrals of matrix valued functions......Page p372.djvu
17.6 Continuous dependence of the eigenvalues......Page p375.djvu
17.7 More on small perturbations......Page p377.djvu
17.8 Spectral radius redux......Page p378.djvu
17.9 Fractional powers......Page p381.djvu
18.1 The equation X - AXB = C......Page p383.djvu
18.2 The Sylvester equation AX - XB = C......Page p385.djvu
18.3 Special classes of solutions......Page p388.djvu
18.4 Riccati equations......Page p390.djvu
18.5 Two lemmas......Page p396.djvu
18.6 An LQR problem......Page p398.djvu
18.7 Bibliographical notes......Page p400.djvu
19 Realization theory......Page p401.djvu
19.1 Minimal realizations......Page p408.djvu
19.2 Stabilizable and detectable realizations......Page p415.djvu
19.3 Reproducing kernel Hilbert spaces......Page p416.djvu
19.4 de Branges spaces......Page p418.djvu
19.5 R_{alpha} invariance......Page p420.djvu
19.6 Factorization of Theta(A)......Page p421.djvu
19.7 Bibliographical notes......Page p425.djvu
20.1 Interlacing......Page p427.djvu
20.2 Sylvester's law of inertia......Page p430.djvu
20.3 Congruence......Page p431.djvu
20.4 Counting positive and negative eigenvalues......Page p433.djvu
20.5 Exploiting continuity......Page p437.djvu
20.6 Geršgorin disks......Page p438.djvu
20.7 The spectral mapping principle......Page p439.djvu
20.8 AX = XB......Page p440.djvu
20.9 Inertia theorems......Page p441.djvu
20.10 An eigenvalue assignment problem......Page p443.djvu
20.11 Bibliographical notes......Page p446.djvu
21.1 Bezoutians......Page p447.djvu
21.2 A derivation of the formula for H_f based on realization......Page p452.djvu
21.3 The Barnett identity......Page p453.djvu
21.4 The main theorem on Bezoutians......Page p455.djvu
21.5 Resultants......Page p457.djvu
21.6 Other directions......Page p461.djvu
21.7 Bezoutians for real polynomials......Page p463.djvu
21.8 Stable polynomials......Page p464.djvu
21.9 Kharitonov's theorem......Page p466.djvu
21.10 Bibliographical notes......Page p467.djvu
22.1 Preliminaries......Page p469.djvu
22.2 Convex functions......Page p471.djvu
22.3 Convex sets in R^n......Page p473.djvu
22.4 Separation theorems in R^n......Page p475.djvu
22.5 Hyperplanes......Page p477.djvu
22.6 Support hyperplanes......Page p479.djvu
22.7 Convex hulls......Page p480.djvu
22.8 Extreme points......Page p482.djvu
22.10 The Minkowski functional......Page p485.djvu
22.11 The Gauss-Lucas theorem......Page p488.djvu
22.12 The numerical range......Page p489.djvu
22.13 Eigenvalues versus numerical range......Page p491.djvu
22.14 The Heinz inequality......Page p492.djvu
22.15 Bibliographical notes......Page p494.djvu
23 Matrices with nonnegative entries......Page p495.djvu
23.1 Perron-Frobenius theory......Page p496.djvu
23.2 Stochastic matrices......Page p503.djvu
23.3 Doubly stochastic matrices......Page p504.djvu
23.4 An inequality of Ky Fan......Page p507.djvu
23.5 The Schur-Horn convexity theorem......Page p509.djvu
23.6 Bibliographical notes......Page p513.djvu
A.1 Convergence of sequences of points......Page p515.djvu
A.3 Convergence of sums......Page p516.djvu
A.4 Sups and infs......Page p517.djvu
A.7 Normed linear spaces......Page p518.djvu
B.1 Power series......Page p521.djvu
B.2 Isolated zeros......Page p523.djvu
B.4 ln(1 - lambda) when |lambda| < 1......Page p525.djvu
B.5 Rouché's theorem......Page p526.djvu
B.7 Laurent expansions......Page p528.djvu
B.8 Partial fraction expansions......Page p529.djvu
Bibliography......Page p531.djvu
Notation Index......Page p535.djvu
Subject Index......Page p537.djvu