Math 631 Linear Algebra

Fall 2002

Prof. Daniel Goldman

Textbook: Matrix Theory by Joel N. Franklin

Grading: Homework (30%), Mid-Term Exam (30%), Final Exam (40%)

Homework Assignments

Tentative Outline:

Determinants

• permutations, definition of the determinant, effect of elementary matrix operations on the determinant (1.1, 1.2, 1.3)
• expansion by minors, matrix arithmetic, the inverse matrix and Crammer’s rule (1.4, 1.5, 1.6)
• determinant of products, implications of a vanishing determinant, derivative of a determinant (1.7, 1.8)

Vector Spaces and Linear Systems

• vector spaces, subspace, linear independence (2.1, 2.2)
• span, basis, dimension (2.3)
• homogeneous solutions, rank (2.4, 2.5)
• solutions of general linear systems (2.6)
• permutation matrices, elementary row operations, LU factorization (7.1, 7.2, 7.3)

Linear Differential Equations

• representation as first-order systems of ODES and handling inhomogeneous terms (3.1, 3.2, 3.3)
• exponential of a matrix (3.4)
• eigensolutions (3.5)

Eigenvalues, Eigenvectors, and Canonical Forms

• eigenvalues and eigenvectors, characteristic polynomial, independence of eigenvectors corresponding to distinct eigenvalues (4.1)
• change of coordinates, similar matrices, diagonalization, invariants (4.2, 4.3)
• Euclidean norm, inner product, Cauchy-Schwarz inequality, unitary and orthogonal matrices, transpose and inverse (4.4)
• orthogonal basis, the Gram-Schmidt orthogonalization process, the QR factorization (4.5)
• principal axes of ellipsoids, quadratic forms (4.6)
• Hermitian and symmetric matrices: real eigenvalues, orthogonal basis of eigenvectors (4.7)
• mass spring systems, positive definite matrices, generalized eigenvalue problem (4.8)
• unitary triangularization, perturbation to distinct eigenvalues, convergence to the zero matrix, Cayley-Hamilton theorem, normal matrices (4.9, 4.10)
• the Jordan canonical form (5.1, 5.2, 5.3)

Variational Principles and Perturbation Theory

• Rayleigh principle (6.1, 6.2)
• minimax principle and inclusion principle (6.3, 6.4)
• criteria for positive definiteness, Hadamard’s inequality (6.5, 6.6)
• Weyl’s inequality, Gershgorin’s theorem (6.7, 6.8)