Title:

# Matrices and Tensors Calculus

Code:MMAT (FEKT MMAT)
Ac.Year:2019/2020
Sem:Summer
Curriculums:
ProgrammeField/
Specialization
YearDuty
IT-MSC-2MBI-Elective
IT-MSC-2MBS-Elective
IT-MSC-2MGM-Elective
IT-MSC-2MIN-Elective
IT-MSC-2MIS-Elective
IT-MSC-2MMI-Elective
IT-MSC-2MMM-Elective
IT-MSC-2MPV-Elective
IT-MSC-2MSK-Elective
Language of Instruction:Czech
Credits:5
Completion:credit+exam (written&verbal)
Type of
instruction:
Hour/semLecturesSeminar
Exercises
Laboratory
Exercises
Computer
Exercises
Other
Hours:2600188
ExamsTestsExercisesLaboratoriesOther
Points:70200010
Guarantor:Kovár Martin, doc. RNDr., Ph.D. (DMAT)
Lecturer:Kovár Martin, doc. RNDr., Ph.D. (DMAT)
Instructor:Hlavičková Irena, Mgr., Ph.D. (DMAT)
Kovár Martin, doc. RNDr., Ph.D. (DMAT)
Faculty:Faculty of Electrical Engineering and Communication BUT
Department:Department of Mathematics FEEC BUT

Learning objectives:
Master the bases of the matrices and tensors calculus and its applications.
Description:
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.
Learning outcomes and competencies:
Mastering basic techniques for solving tasks and problems from the matrices and tensors calculus and its applications.
Syllabus of lectures:

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

1. Havel, V., Holenda, J.: Lineární algebra, SNTL, Praha 1984 (in Czech).
2. Hrůza, B., Mrhačová, H.: Cvičení z algebry a geometrie. Ediční stř. VUT 1993, skriptum (in Czech).
3. Schmidtmayer J.: Maticový počet a jeho použití, SNTL, Praha, 1967 (in Czech).
4. Boček, L.: Tenzorový počet, SNTL Praha 1976 (in Czech).
5. Angot A.: Užitá matematika pro elektroinženýry, SNTL, Praha 1960 (in Czech).
6. Kolman, B.: Elementary Linear Algebra, Macmillan Publ. Comp., New York 1986.
7. Kolman, B.: Introductory Linear Algebra, Macmillan Publ. Comp., New York 1991.
8. Gantmacher, F. R.: The Theory of Matrices, Chelsea Publ. Comp., New York 1960.
9. Demlová, M., Nagy, J.: Algebra, STNL, Praha 1982 (in Czech).
10. Plesník J., Dupačová, J., Vlach M.: Lineárne programovanie, Alfa, Bratislava , 1990 (in Slovak).
11. Mac Lane, S., Birkhoff, G.: Algebra, Alfa, Bratislava, 1974 (in Slovak).
12. Mac Lane, S., Birkhoff, G.: Prehľad modernej algebry, Alfa, Bratislava, 1979 (in Slovak).
13. Krupka D., Musilová J.: Lineární a multilineární algebra, Skriptum Př. f. MU, SPN, Praha, 1989 (in Czech).
14. Procházka, L. a kol.: Algebra, Academia, Praha, 1990 (in Czech).
Halliday, D., Resnik, R., Walker, J.: Fyzika, Vutium, Brno, 2000 (in Czech).
15. Halliday D., Resnik R., Walker J., Fyzika, Vutium, Brno, 2000. (in Czech).
16. Crandal, R. E.: Mathematica for the Sciences, Addison-Wesley, Redwood City, 1991.
17. Davis, H. T., Thomson, K. T.: Linear Algebra and Linear Operators in Engineering, Academic Press, San Diego, 2007.
18. Mannuci, M. A., Yanofsky, N. S.: Quantum Computing For Computer Scientists, Cambridge University Press, Cabridge, 2008.
19. Nahara, M., Ohmi, T.: Quantum Computing: From Linear Algebra to Physical Realizations, CRC Press, Boca Raton, 2008.
20. Griffiths, D.: Introduction to Elementary Particles, Wiley WCH, Weinheim, 2009.
Controlled instruction:
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.
Progress assessment:
Requirements for completion of a course are specified by a regulation issued by the lecturer responsible for the course and updated for every.