Course detail

Matrices and Tensors Calculus

FIT-MMATAcad. year: 2019/2020

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

Not applicable.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

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.
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

Not applicable.

Assesment methods and criteria linked to learning outcomes

Requirements for completion of a course are specified by a regulation issued by the lecturer responsible
for the course and updated for every.

Language of instruction

Czech

Work placements

Not applicable.

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.

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

  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.
  7. Bilinear and quadratic forms. Definitness of quadratic forms.
  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.

seminars in computer labs

18 hours, compulsory

Teacher / Lecturer

Projects

8 hours, compulsory

Teacher / Lecturer