Course details

Matrices and Tensors Calculus

MMAT Acad. year 2016/2017 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 (written+oral)

Time span

  • 26 hrs lectures
  • 18 hrs pc labs
  • 8 hrs projects

Assessment points

  • 100 pts final exam

Department

Department of Mathematics (UMAT)

Learning objectives

Master the foundations of the matrices and tensors calculus and its applications (including the basics of matrix quantum mechanics and informatics).

Prerequisite knowledge and skills

Discrete mathematics - IDA

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.

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.
  • 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.
  • 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

  1. Definice matice, základní pojmy. Transponování matic.
  2. Determinant čtvercové komplexní matice.
  3. Operace s maticemi, speciální tvary matic. Inverzní matice.
  4. Použití matic k řešení soustav lineárních algebraických rovnic.
  5. Lineární, bilineární a kvadratické formy. Definitnost kvadratických forem.
  6. Spektrální vlastnosti matic.
  7. Lineární prostor, báze, dimenze.
  8. Lineární transformace souřadnic vektoru.
  9. Kovariantní a kontravariantní souřadnice vektoru.
  10. Definice tenzoru.
  11. Tenzor kovariantní, kontravariantní, smíšený.
  12. Operace s tenzory.
  13. Symetrie a antisymetrie tenzorů druhého řádu.

Syllabus of computer exercises

Ověřování poznatků z přednášek a doporučené literatury.

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