Solutions Manual to accompany Fundamentals of Matrix Analysis with Applications

1 Systems of Linear Algebraic Equations 1

1.1 Linear Algebraic Equations 1

1.2 Matrix Representation of Linear Systems and the Gauss-Jordan Algorithm 12

1.3 The Complete Gauss Elimination Algorithm 17

1.4 Echelon Form and Rank 26

1.5 Computational Considerations 34

2 Matrix Algebra 39

2.1 Matrix Multiplication 39

2.2 Some Useful Applications of Matrix Operators 48

2.3 The Inverse and the Transpose 54

2.4 Determinants 61

2.5 Three Important Determinant Rules 68

PART I REVIEW PROBLEMS FOR PART I 79

3 Vector Spaces 89

3.1 General Spaces, Subspaces, and Spans 89

3.2 Linear Dependence 93

3.3 Bases, Dimension, and Rank 97

4 Orthogonality 105

4.1 Orthogonal Vectors and the Gram-Schmidt Algorithm 105

4.2 Orthogonal Matrices 115

4.3 Least Squares 123

4.4 Function Spaces 133

PART II REVIEW PROBLEMS FOR PART II 140

5 Eigenvectors and Eigenvalues 144

5.1 Eigenvector Basics 144

5.2 Calculating Eigenvalues and Eigenvectors 155

5.3 Symmetric and Hermitian Matrices 168

6 Similarity 181

6.1 Similarity Transformations and Diagonalizability 181

6.2 Principal Axes and Normal Modes 189

6.3 Schur Decomposition and Its Implications 198

6.4 The Singular Value Decomposition 212

6.5 The Power Method and the QR Algorithm 217

7 Linear Systems of Differential Equations 221

7.1 First-Order Linear Systems 221

7.2 The Matrix Exponential Function 230

7.3 The Jordan Normal Form 236

7.4 Matrix Exponentiation via Generalized Eigenvectors 246

PART III REVIEW PROBLEMS FOR PART III 255