Course details

# Matrices and Tensors Calculus

MMAT FEEC BUT MPC-MAT Acad. year 2020/2021 Summer semester 5 credits

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.

Guarantor

Language of instruction

Completion

Time span

Assessment points

Department

Lecturer

Instructor

Subject specific learning outcomes and competences

Mastering basic techniques for solving tasks and problems from the matrices and tensors calculus and its applications.

Learning objectives

Master the bases of the matrices and tensors calculus and its applications.

Study literature

- Havel, V., Holenda, J.: Lineární algebra, SNTL, Praha 1984 (in Czech).
- Hrůza, B., Mrhačová, H.: Cvičení z algebry a geometrie. Ediční stř. VUT 1993, skriptum (in Czech).
- Schmidtmayer J.: Maticový počet a jeho použití, SNTL, Praha, 1967 (in Czech).
- Boček, L.: Tenzorový počet, SNTL Praha 1976 (in Czech).
- Angot A.: Užitá matematika pro elektroinženýry, SNTL, Praha 1960 (in Czech).
- Kolman, B.: Elementary Linear Algebra, Macmillan Publ. Comp., New York 1986.
- Kolman, B.: Introductory Linear Algebra, Macmillan Publ. Comp., New York 1991.
- Gantmacher, F. R.: The Theory of Matrices, Chelsea Publ. Comp., New York 1960.
- Demlová, M., Nagy, J.: Algebra, STNL, Praha 1982 (in Czech).
- Plesník J., Dupačová, J., Vlach M.: Lineárne programovanie, Alfa, Bratislava , 1990 (in Slovak).
- Mac Lane, S., Birkhoff, G.: Algebra, Alfa, Bratislava, 1974 (in Slovak).
- Mac Lane, S., Birkhoff, G.: Prehľad modernej algebry, Alfa, Bratislava, 1979 (in Slovak).
- Krupka D., Musilová J.: Lineární a multilineární algebra, Skriptum Př. f. MU, SPN, Praha, 1989 (in Czech).
- Procházka, L. a kol.: Algebra, Academia, Praha, 1990 (in Czech).

Halliday, D., Resnik, R., Walker, J.: Fyzika, Vutium, Brno, 2000 (in Czech). - Halliday D., Resnik R., Walker J., Fyzika, Vutium, Brno, 2000. (in Czech).
- Crandal, R. E.: Mathematica for the Sciences, Addison-Wesley, Redwood City, 1991.
- Davis, H. T., Thomson, K. T.: Linear Algebra and Linear Operators in Engineering, Academic Press, San Diego, 2007.
- Mannuci, M. A., Yanofsky, N. S.: Quantum Computing For Computer Scientists, Cambridge University Press, Cabridge, 2008.
- Nahara, M., Ohmi, T.: Quantum Computing: From Linear Algebra to Physical Realizations, CRC Press, Boca Raton, 2008.
- Griffiths, D.: Introduction to Elementary Particles, Wiley WCH, Weinheim, 2009.

Syllabus of lectures

- 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.

Syllabus - others, projects and individual work of students

Two projects on selected topics in applied mathematics, each per 5 points.

Progress assessment

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.

Controlled instruction

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

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.

Schedule

Day | Type | Weeks | Room | Start | End | Lect.grp | Groups | Info |
---|---|---|---|---|---|---|---|---|

Mon | lecture | lectures | T8/010 | 11:00 | 12:50 | 1MIT 2MIT | xx | Kovár |

Thu | exercise | lectures | T8/522 | 10:00 | 11:50 | 1MIT 2MIT | xx | Kovár |

Thu | other | lectures | T8/522 | 10:00 | 11:50 | 1MIT 2MIT | xx | Kovár |

Thu | exercise | lectures | T8/522 | 12:00 | 13:50 | 1MIT 2MIT | xx | Hlavičková |

Thu | other | lectures | T8/522 | 12:00 | 13:50 | 1MIT 2MIT | xx | Hlavičková |

Thu | exercise | lectures | T8/522 | 14:00 | 15:50 | 1MIT 2MIT | xx | Hlavičková |

Thu | other | lectures | T8/522 | 14:00 | 15:50 | 1MIT 2MIT | xx | Hlavičková |

Course inclusion in study plans

- Programme IT-MSC-2, field MBI, MBS, MGM, MIN, MIS, MMI, MMM, MPV, MSK, any year of study, Elective
- Programme MITAI, specialisation NADE, NBIO, NCPS, NEMB, NGRI, NIDE, NISD, NISY, NMAL, NMAT, NNET, NSEC, NSEN, NSPE, NVER, NVIZ, any year of study, Elective
- Programme MITAI, specialisation NHPC, 1st year of study, Compulsory