Course details

Matrices and Tensors Calculus

MMAT FEKT MMAT Acad. year 2019/2020 Summer semester 5 credits

Current academic year

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

Kovár Martin, doc. RNDr., Ph.D. (UMAT)

Language of instruction

Czech

Completion

Credit+Examination

Time span

  • 26 hrs lectures
  • 18 hrs exercises

Department

Department of Mathematics (UMAT)

Lecturer

Kovár Martin, doc. RNDr., Ph.D. (UMAT)

Instructor

Hlavičková Irena, Mgr., Ph.D. (UMAT)
Kovár Martin, doc. RNDr., Ph.D. (UMAT)

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.

Prerequisite knowledge and skills

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.

Study literature

  • Gantmacher, F. R., The Theory of Matrices, Chelsea Publ. Comp., New York 1960.
  • 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.
  • 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.

Fundamental 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.
  • Demlová, M., Nagy, J., Algebra, STNL, Praha 1982.
  • Krupka D., Musilová J., Lineární a multilineární algebra, Skriptum Př. f. MU, SPN, Praha, 1989.

Syllabus of lectures

Definition of matrix, fundamental notion. Transposition of matrices.
Determinant of quadratic complex matrix.
Operations with matrices. Special types of matrices. Inverse matrix.
Matrix solutions of linear algebraic equations.
Linear, bilinear and quadratic forms. Definite of quadratics forms.
Spectral attributes of matrices.
Linear space, dimension.
Linear transform of coordinates of vector.
Covariant and contravariant coordinates of vector.
Definition of tensor.
Covariant, contravariant and mixed tensor.
Operation with tensors.
Symmetry and antisymmetry of tensors of second order.

Syllabus of numerical exercises

Operations with matrices. Inverse matrices. Using matrices for solving systems of linear algebraic equations.
Spectral properties of matrices.
Operations with tensors.

Progress assessment

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.

Teaching methods and criteria

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

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.

Course inclusion in study plans

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