Math 570: Matrix Analysis
Special classes of matrices, canonical forms, matrix and vector norms, localization of eigenvalues, matrix functions, applications.
Desired Learning Outcomes
Math 570 is a one semester course on matrix analysis.
Math 313 or 302 or equivalent and Math 112, 113, 314.
Minimal learning outcomes
Outlined below are the minimal learning outcomes which all students in Math 570 should understand. As evidence of that understanding, students should be able to demonstrate mastery of relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.
Matrix arithmetic and Linear transformations, The theory of determinants including all proofs of their properties, Rank of a matrix and elementary matrices, Spectral theory, Shur's theorem, Principal invariants of trace and determinant, Quadratic forms and second derivative test, Gerschgorin's theorem, Abstract vector spaces and general fields, Axioms, Subspaces and bases, Applications to general fields, Linear transformations, Matrix of a linear transformation, Rotations, Eigenvalues and eigenvectors of linear transformations, Jordan Cannonical form and applications, Cayley Hamilton theorem, Markov chains and migration processes, Regular Markov matrices, Absorbing states and gambler's ruin, Inner product spaces, Gramm Schmidt process, Tensor product of vectors, Least squares, Fredholm alternative, Determinants and volume, Self adjoint operators, Simultaneous diagonalization, Spectral theory of self adjoint operators, Positive and negative linear transformations, Fractional powers, Polar decompositions and applications, Singular value decomposition, The Frobenius norm and approximation in this norm, Least squares and the Moore Penrose inverse, Norms for finite dimensional vector spaces, The p norms, The condition number, The spectral radius, Sequences and series of linear operators, Functions of linear transformations, Iterative methods for solutions of linear systems,
Possible textbooks for this course include (but are not limited to):
Horn and Johnson, Friedberg, Insel and Spence, or equivalent.
Numerical methods for finding eigenvalues, Power methods, The QR algorithm, Rational canonical form,
Courses for which this course is prerequisite