Course detail

Matrices and Tensors Calculus

FEKT-MKC-MATAcad. year: 2024/2025

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

5

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

Demlová, M., Nagy, J., Algebra, STNL, Praha 1982. (CS)
Havel V., Holenda J.: Lineární algebra, SNTL, Praha 1984. (CS)
Hrůza B., Mrhačová H.: Cvičení z algebry a geometrie. Ediční stř. VUT 1993, skriptum (CS)
Kolman, B., Elementary Linear Algebra, Macmillan Publ. Comp., New York 1986. (EN)
Kolman, B., Introductory Linear Algebra, Macmillan Publ. Comp., New York 1991. (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 reading

Boček L.: Tenzorový počet, SNTL Praha 1976. (CS)
Crandal R. E., Mathematica for the Sciences, Addison-Wesley, Redwood City, 1991. (EN)
Davis H. T., Thomson K. T., Linear Algebra and Linear Operators in Engineering, Academic Press, San Diego, 2007. (EN)
Gantmacher, F. R., The Theory of Matrices, Chelsea Publ. Comp., New York 1960. (EN)
Krupka D., Musilová J., Lineární a multilineární algebra, Skriptum Př. f. MU, SPN, Praha, 1989. (CS)
Mannuci M. A., Yanofsky N. S., Quantum Computing For Computer Scientists, Cambridge University Press, Cabridge, 2008. (EN)
Nahara M., Ohmi T., Quantum Computing: From Linear Algebra to Physical Realizations, CRC Press, Boca Raton, 2008. (EN)
Plesník J., Dupačová J., Vlach M., Lineárne programovanie, Alfa, Bratislava, 1990. (CS)
Schmidtmayer J.: Maticový počet a jeho použití, SNTL, Praha, 1967. (CS)

Classification of course in study plans

  • Programme MPC-EVM 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-TIT Master's 1 year of study, summer semester, compulsory-optional

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Martin Kovár, Ph.D.

Exercise in computer lab

18 hod., compulsory

Teacher / Lecturer

doc. RNDr. Martin Kovár, Ph.D.

Project

8 hod., optionally

Teacher / Lecturer

doc. RNDr. Martin Kovár, Ph.D.

Syllabus

Dva projekty na vybraná témata z aplikované matematiky, každý po 5 bodech.