## Math 464/514 - Applied Matrix Theory, Fall 2017

This course along with Math 504 prepares graduate students for the Numerical Analysis Qualifying Exam.

Time: Tue & Thur 9:30-10:45 a.m., Centennial Engineering Center 1028
Instructor Info: Anna Skripka, skripka [at] math [dot] unm [dot] edu

Prerequisite: Math 314 or 321

Lecture notes by Jens Lorenz
Textbook for supplementary reading: Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000.

Office hours in SMLC 212: Tue and Thur 11 a.m. - noon, and also by appointment

Grades will be based on homework.

Homework is to be posted on UNM Learn.

### Schedule

Aug 22, 24:      Review. Gaussian elimination without interchanges. LU factorization.
(Lecture notes 1.1, 1.4. Textbook 1.2, 1.3, 1.5, 3.2, 3.5, 3.6, 3.7, 3.9, 3.10.)
HW1 due August 31 is posted on UNM Learn.

Aug 29, 31:      Gaussian elimination with partial pivoting. Factorization PA=LU and its applications.
(Lecture notes 1.5-1.8, 1.3. Textbook 3.8, 1.4.)
HW2 due September 7 is posted on UNM Learn.

Sep 5, 7:      Norms and inner products. Conditioning of linear systems.
(Lecture notes 2.1-3.1. Textbook 3.10, 5.1, 5.2, 5.3.)
HW3 due September 14

Sep 12, 14:      Fundamental subspaces of a rectangular matrix. Orthogonal complements.
(Lecture notes 4.1-4.5. Textbook 4.1-4.4, 5.11.)
HW4 due September 21

Sep 19, 21:      Direct sums, projections. Least squares problems and their applications.
(Lecture notes 5.1-5.4, 7.1, 7.2, 3.2. Textbook 5.9, 5.13.)
HW5 due September 28

Sep 26, 28:      Gram-Schmidt orthogonalization, QR factorization. Householder reduction.
(Lecture notes 7.3-7.6. Textbook 5.5, 5.6, 5.7.)
HW6 due October 10

Oct 3, 5:      Singular value decomposition. Applications to least squares problems. Pseudoinverse.
(Lecture notes 8.1-8.3. Textbook 5.12.)

Oct 10:      Frobenius norm. SVD, rank, condition number. Application to information storage.
(Lecture notes 8.4, 8.5, 2.2. Textbook 5.12.)
HW7 due October 19

Oct 17, 19:      Sign of a permutation. Determinant.
(Lecture notes 9.1, 9.3. Textbook 6.1.)

Oct 24, 26:      Properties and applications of determinants. Connection to volumes.
(Lecture notes 9.4, 9.2. Textbook 6.1, 6.2.)
HW8 due November 2

Oct 31, Nov 2:      Eigenvalues and eigenvectors. Schur decomposition. Diagonalization of a normal matrix.
(Lecture notes 9.5, 9.7, 10.1-10.5. Textbook 7.1, 7.2, 7.5.)
HW9 due November 9

Nov 7, 9:      Matrix exponential. Systems of differential equations. Stability, separation of variables.
(Lecture notes 10.6, 11.1, 11.2. Textbook 7.3, 7.4.)
HW10 due November 16

Nov 14, 16:      Jordan normal form.
(Lecture notes 12.1-12.5. Textbook 7.7, 7.8.)

Nov 21:      Limiting behaviour of matrix powers and matrix exponential.
(Lecture notes 12.6, 12.7.)
HW11 due November 30

Nov 28, 30:      Minimal polynomials. Krylov subspace methods. GMRES. Gradient method.
(Lecture notes 15.1-15.4. Textbook 7.11.)

Dec 5, 7:      Conjugate gradient method. Review.
(Textbook 7.11.)

Optional Final Exam: Dec 12 (Tuesday), 12:30 - 2:30 pm, DSH 232.

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