Course detail

Matrices and Tensors Calculus

FEKT-MPC-MATAcad. year: 2023/2024

Matrices as algebraic structure. Matrix operations. Determinant. Matrices in systems of linear algebraic equations. Vector space, its basis and dimension. Coordinates and their transformation. Sum and intersection of vector spaces. Linear mapping of vector spaces and its matrix representation. Inner (dot) product, orthogonal projection and the best approximation element. Eigenvalues and eigenvectors. Spectral properties of (especially Hermitian) matrices. Bilinear and quadratic forms. Definitness of quadratic forms. Linear forms and tensors. Verious types of coordinates. Covariant, contravariant and mixed tensors. Tensor operations. Tensor and wedge products.Antilinear forms. Matrix formulation of quantum. Dirac notation. Bra and Ket vectors. Wave packets as vectors. Hermitian linear operator. Schrodinger equation. Uncertainty Principle and Heisenberg relation. Multi-qubit systems and quantum entaglement. Einstein-Podolsky-Rosen experiment-paradox. Quantum calculations. Density matrix. Quantum teleportation.

Language of instruction

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

doc. RNDr. Martin Kovár, Ph.D.

Department

Department of Mathematics (UMAT)

Entry knowledge

The knowledge of the content of the subject Matematika 1 is required. The previous attendance to the subject Matematický seminář is warmly recommended.

Rules for evaluation and completion of the course

The semester examination is rated at a maximum of 70 points. It is possible to get a maximum of 30 points in practices, 20 of which are for written tests and 10 points for 2 project solutions, 5 points of each.
The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year.

Aims

Master the bases of the matrices and tensors calculus and its applications.
The student will brush up and improve his skills in

- solving the systems of linear equations
- calculating determinants of higher order using various methods
- using various matrix operations

The student wil further learn up to

- find the basis and dimension of a vector space
- express the vectors in various bases and calculate their coordinates
- calculate the intersection and sum of vector spaces
- find the ortohogonal projection of a vector into a vector subspace
- find the orthogonal complement of a vector subspace
- calculate the eigenvalues and the eigenvectors of a square matrix
- find the spectral representation of a Hermitian matrix
- determine the type of a conic section or a quadric
- classify a quadratic form with respect to its definiteness
- express tensors in various types of bases
- calculate various types of tensor products
- use the matrix representation for selected quantum quantities and calculations

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Kolman, B., Elementary Linear Algebra, Macmillan Publ. Comp., New York, ISBN 978-0029463703, 1986. (EN)
Kolman, B., Hill, D. R., Introductory Linear Algebra, Pearson, New York, 978-8131723227, 2008. (EN)
Kovár, M., Maticový a tenzorový počet, Skriptum, Brno, 2013, 220s. (CS)
Kovár, M., Selected Topics on Multilinear Algebra with Applications, Skriptum, Brno, 2015, 141s. (EN)

Recommended literature

Crandal, R. E., Mathematica for the Sciences, Addison-Wesley, Redwood City, ISBN 978-0201510010, 1991. (EN)
Davis, H. T., Thomson K. T., Linear Algebra and Linear Operators in Engineering, Academic Press, San Diego, ISBN 978-0122063497, 2007. (EN)
Demlová, M., Nagy, J., Algebra, STNL, Praha 1982. (CS)
Havel, V., Holenda J.: Lineární algebra, SNTL, Praha 1984. (CS)

Elearning

eLearning: currently opened course

Classification of course in study plans

Programme MPC-TIT Master's 1 year of study, summer semester, compulsory-optional
Programme MPC-SVE Master's 1 year of study, summer semester, compulsory-optional
Programme MPC-MEL Master's 1 year of study, summer semester, compulsory-optional
Programme MPC-IBE Master's 1 year of study, summer semester, compulsory
Programme MPC-EVM Master's 1 year of study, summer semester, compulsory-optional
Programme MPC-EKT Master's 1 year of study, summer semester, compulsory-optional
Programme MPC-EEN Master's 0 year of study, summer semester, elective
Programme MPC-EAK Master's 0 year of study, summer semester, elective
Programme MPC-BIO Master's 1 year of study, summer semester, compulsory-optional
Programme MPC-AUD Master's
specialization AUDM-TECH , 1 year of study, summer semester, compulsory-optional
specialization AUDM-ZVUK , 1 year of study, summer semester, compulsory-optional
Programme IT-MSC-2 Master's
branch MBS , 0 year of study, summer semester, elective
branch MPV , 0 year of study, summer semester, elective
branch MIS , 0 year of study, summer semester, elective
branch MIN , 0 year of study, summer semester, elective
branch MGM , 0 year of study, summer semester, elective
branch MBI , 0 year of study, summer semester, elective
branch MSK , 0 year of study, summer semester, elective
branch MMM , 0 year of study, summer semester, elective
Programme MITAI Master's
specialization NISY , 0 year of study, summer semester, elective
specialization NSPE , 0 year of study, summer semester, elective
specialization NBIO , 0 year of study, summer semester, elective
specialization NSEN , 0 year of study, summer semester, elective
specialization NVIZ , 0 year of study, summer semester, elective
specialization NGRI , 0 year of study, summer semester, elective
specialization NADE , 0 year of study, summer semester, elective
specialization NISD , 0 year of study, summer semester, elective
specialization NMAT , 0 year of study, summer semester, elective
specialization NSEC , 0 year of study, summer semester, elective
specialization NISY up to 2020/21 , 0 year of study, summer semester, elective
specialization NCPS , 0 year of study, summer semester, elective
specialization NHPC , 1 year of study, summer semester, compulsory
specialization NNET , 0 year of study, summer semester, elective
specialization NMAL , 0 year of study, summer semester, elective
specialization NVER , 0 year of study, summer semester, elective
specialization NIDE , 0 year of study, summer semester, elective
specialization NEMB , 0 year of study, summer semester, elective
specialization NEMB up to 2021/22 , 0 year of study, summer semester, elective