Numerical Linear Algebra
- define and describe fundamental concepts such as matrix norms, singular and eigen values and vectors
- recognize types of matrices, such as ortogonal, unitary, symetric, Hermitian, normal and positive definite matrices
- describe Jordan and Schure form of the matrix and a notion of a diagonalizable matrix
- apply matrix transformation in order to transform a matrix into triangular, Hessenberg and tridiagonal form
- derive and utilize SVD and QR factorization of the matrix for efficiently solving problems in practice
- analyze and employ iterative methods for linear algebraic systems
- explain and employ iterative algorithms for computing eigenvalues
- Relate the quality of obtained numerical solution to derived theoretical results
Forms of Teaching
Week by Week Schedule
- Symmetric and i Hermitian (self-adjoint) matrices; Gramm-Schmidt's orthogonalization; Orthogonal and unitary matrices.
- Diagonalization of a matrix; Spectral theorem for symmetric matrices; Nilpotent matrices; Jordan cells; Minimal polynomial.
- Nilpotent matrices; Jordan cells; Minimal polynomial; Positively (negatively) semidefinite matrices; Positively (negatively) definite matrices; Indefinite matrices.
- Quadratic forms and their diagonalization; Definite and indefinite quadratic forms; Signature; Minimum, maximum and saddle points of a quadratic form.
- Operator norms; The space L(X,Y); Matrix norm; Convergence of matrices; Series of matrices; Neumann series; Spectral radius and spectral norm.
- Projectors; QR factorization; HouseHolder triangularization and its stability.
- Singular Value Decomposition and its application; Algorithms for the SVD.
- Midterm exam.
- Jacobi, Gauss-Seidel and Relaxation Methods; Convergence Results for Jacobi, Gauss-Seidel and Relaxation Methods.
- Symmetric Form of the Gauss-Seidel and SOR Methods; Implementation Issues.
- Conjugate Gradient Method and other Krylov Subspace Iterations; Applications: Analysis of an Electric Network; Finite Difference Analysis of Beam Bending.
- Geometrical Location of the Eigenvalues; Stability and Conditioning Analysis; The Power Method; Inverse Iteration.
- The QR Method; Hessenberg Reduction; Tridiagonal and Bidiagonal Reductions; QR Iterations with Implicit Shifts.
- Methods for Eigenvalues of Symmetric Matrices (The Jacobi's Method; Tridiagonal QR Iteration, Rayleigh Quotient Iteration); Applications: Free Dynamic Vibration of a Bridge.
- Final exam.
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(.), A. Aglić Aljinović, N. Elezović, D. Žubrinić, Linearna algebra, Element, Zagreb, 2011.,
(.), Z. Drmač i ostali, Numerička analiza (predavanja i vježbe), Zagreb, 2003. https://web.math.pmf.unizg.hr/~rogina/2001096/num_anal.pdf),
(.), N. Truhar, Numerička linearna algebra, Osijek, 2012. http://www.mathos.unios.hr/nla/NLA.pdf,
(.), G. H. Golub i C. F. Van Loan: Matrix Computations, 3rd Edition, John Hopkins University Press, Baltimore, Maryland, 1996.,
(.), L.N. Trefethen, D. Bau: Numerical Linear Algebra, SIAM, 1997.,
L0 English Level
13 Laboratory exercises
0 Project laboratory