Course detail

# Matrices and Tensors Calculus

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 of the course unit

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

Prerequisites

The knowledge of the content of the subject BMA1 Matematika 1 is required. The previous attendance to the subject BMAS Matematický seminář is warmly recommended.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Literature

Havel V., Holenda J.: Lineární algebra, SNTL, Praha 1984.
Hrůza B., Mrhačová H.: Cvičení z algebry a geometrie. Ediční stř. VUT 1993, skriptum
Schmidtmayer J.: Maticový počet a jeho použití, SNTL, Praha, 1967.
Boček L.: Tenzorový počet, SNTL Praha 1976.
Angot A.: Užitá matematika pro elektroinženýry, SNTL, Praha 1960.
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.
Plesník J., Dupačová J., Vlach M., Lineárne programovanie, Alfa, Bratislava , 1990.
Mac Lane S., Birkhoff G., Algebra, Alfa, Bratislava, 1974.
Mac Lane S., Birkhoff G., Prehľad modernej algebry, Alfa, Bratislava, 1979.
Krupka D., Musilová J., Lineární a multilineární algebra, Skriptum Př. f. MU, SPN, Praha, 1989.
Procházka L. a kol., Algebra, Academia, Praha, 1990.
Halliday D., Resnik R., Walker J., Fyzika, Vutium, Brno, 2000.
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.

Planned learning activities and teaching methods

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

Assesment methods and criteria linked to learning outcomes

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.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

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.

Aims

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

Specification of controlled education, way of implementation and compensation for absences

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.

Classification of course in study plans

• Programme IT-MGR-2 Master's

branch MBI , any year of study, summer semester, 5 credits, elective
branch MPV , any year of study, summer semester, 5 credits, elective
branch MGM , any year of study, summer semester, 5 credits, elective
branch MSK , any year of study, summer semester, 5 credits, elective
branch MIS , any year of study, summer semester, 5 credits, elective
branch MBS , any year of study, summer semester, 5 credits, elective
branch MIN , any year of study, summer semester, 5 credits, elective
branch MMI , any year of study, summer semester, 5 credits, elective
branch MMM , any year of study, summer semester, 5 credits, elective

• Programme AUDIO-P Master's

branch P-AUD , 1. year of study, summer semester, 5 credits, optional interdisciplinary

• Programme EEKR-M1 Master's

branch M1-TIT , 1. year of study, summer semester, 5 credits, theoretical subject
branch M1-KAM , 1. year of study, summer semester, 5 credits, theoretical subject
branch M1-EVM , 1. year of study, summer semester, 5 credits, theoretical subject
branch M1-EST , 1. year of study, summer semester, 5 credits, theoretical subject
branch M1-SVE , 1. year of study, summer semester, 5 credits, theoretical subject
branch M1-EEN , 1. year of study, summer semester, 5 credits, theoretical subject

• Programme IBEP-V Master's

branch V-IBP , 1. year of study, summer semester, 5 credits, compulsory

• Programme AUDIO-P Master's

branch P-AUD , 2. year of study, summer semester, 5 credits, optional interdisciplinary

• Programme EEKR-CZV lifelong learning

branch ET-CZV , 1. year of study, summer semester, 5 credits, theoretical subject

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Computer-assisted exercise

18 hours, compulsory

Teacher / Lecturer

The other activities

8 hours, compulsory

Teacher / Lecturer