vi Contents 2.7. The Gram–Schmidt Process and QR Factorization 33 2.8. Linear Functionals and the Dual Space 35 Exercises 36 Chapter 3. Eigenvalues, Eigenvectors, Diagonalization, and Triangularization 39 3.1. Eigenvalues 39 3.2. Algebraic and Geometric Multiplicity 40 3.3. Diagonalizability 41 3.4. A Triangularization Theorem 44 3.5. The Gerˇ sgorin Circle Theorem 45 3.6. More about the Characteristic Polynomial 46 3.7. Eigenvalues of AB and BA 48 Exercises 48 Chapter 4. The Jordan and Weyr Canonical Forms 51 4.1. A Theorem of Sylvester and Reduction to Block Diagonal Form 53 4.2. Nilpotent Matrices 57 4.3. The Jordan Form of a General Matrix 63 4.4. The Cayley–Hamilton Theorem and the Minimal Polynomial 64 4.5. Weyr Normal Form 67 Exercises 74 Chapter 5. Unitary Similarity and Normal Matrices 77 5.1. Unitary Similarity 77 5.2. Normal Matrices—the Spectral Theorem 78 5.3. More about Normal Matrices 81 5.4. Conditions for Unitary Similarity 84 Exercises 86 Chapter 6. Hermitian Matrices 89 6.1. Conjugate Bilinear Forms 89 6.2. Properties of Hermitian Matrices and Inertia 91 6.3. The Rayleigh–Ritz Ratio and the Courant–Fischer Theorem 94 6.4. Cauchy’s Interlacing Theorem and Other Eigenvalue Inequalities 97 6.5. Positive Definite Matrices 99 6.6. Simultaneous Row and Column Operations 102 6.7. Hadamard’s Determinant Inequality 105 6.8. Polar Factorization and Singular Value Decomposition 106 Exercises 109 Chapter 7. Vector and Matrix Norms 113 7.1. Vector Norms 113

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