Title:  Matrices and Tensors Calculus 

Code:  MMAT (FEKT MMAT) 

Ac.Year:  2017/2018 

Term:  Summer 

Curriculums:  

Language:  Czech 

Credits:  5 

Completion:  accreditation+exam (written&verbal) 

Type of instruction:  Hour/sem  Lectures  Sem. Exercises  Lab. exercises  Comp. exercises  Other 

Hours:  26  0  0  18  8 

 Examination  Tests  Exercises  Laboratories  Other 

Points:  70  20  0  0  10 



Guarantee:  Kovár Martin, doc. RNDr., Ph.D., DMAT 

Lecturer:  Kovár Martin, doc. RNDr., Ph.D., DMAT 
Instructor:  Kovár Martin, doc. RNDr., Ph.D., DMAT 

Faculty:  Faculty of Electrical Engineering and Communication BUT 

Department:  Department of Mathematics FEEC BUT 

 Learning objectives: 

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

  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. Multiqubit systems and
quantum entaglement. EinsteinPodolskyRosen experimentparadox. Quantum
calculations. Density matrix. Quantum teleportation.  Learning outcomes and competences: 

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

Multiqubit systems and quantum entaglement. EinsteinPodolskyRosen experimentparadox.

Quantum calculations. Density matrix. Quantum teleportation.
 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, AddisonWesley, 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.
 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.  Progress assessment: 

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