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.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Learning outcomes of the course unit

The student will brush up and improve his skills in

- solving the systems of linear equations
- calculating determinants of higher order using various methods
- using various matrix operations

The student wil further learn up to

- find the basis and dimension of a vector space
- express the vectors in various bases and calculate their coordinates
- calculate the intersection and sum of vector spaces
- find the ortohogonal projection of a vector into a vector subspace
- find the orthogonal complement of a vector subspace
- calculate the eigenvalues and the eigenvectors of a square matrix
- find the spectral representation of a Hermitian matrix
- determine the type of a conic section or a quadric
- classify a quadratic form with respect to its definiteness
- express tensors in various types of bases
- calculate various types of tensor products
- use the matrix representation for selected quantum quantities and calculations

Prerequisites

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

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the 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.

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.

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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Demlová, M., Nagy, J., Algebra, STNL, Praha 1982. (CS)
Havel V., Holenda J.: Lineární algebra, SNTL, Praha 1984. (CS)
Hrůza B., Mrhačová H.: Cvičení z algebry a geometrie. Ediční stř. VUT 1993, skriptum (CS)
Kolman, B., Elementary Linear Algebra, Macmillan Publ. Comp., New York 1986. (EN)
Kolman, B., Introductory Linear Algebra, Macmillan Publ. Comp., New York 1991. (EN)
Kovár, M., Maticový a tenzorový počet, Skriptum, Brno, 2013, 220s. (CS)
Kovár, M., Selected Topics on Multilinear Algebra with Applications, Skriptum, Brno, 2015, 141s. (EN)

Boček L.: Tenzorový počet, SNTL Praha 1976. (CS)
Crandal R. E., Mathematica for the Sciences, Addison-Wesley, Redwood City, 1991. (EN)
Davis H. T., Thomson K. T., Linear Algebra and Linear Operators in Engineering, Academic Press, San Diego, 2007. (EN)
Gantmacher, F. R., The Theory of Matrices, Chelsea Publ. Comp., New York 1960. (EN)
Krupka D., Musilová J., Lineární a multilineární algebra, Skriptum Př. f. MU, SPN, Praha, 1989. (CS)
Mannuci M. A., Yanofsky N. S., Quantum Computing For Computer Scientists, Cambridge University Press, Cabridge, 2008. (EN)
Nahara M., Ohmi T., Quantum Computing: From Linear Algebra to Physical Realizations, CRC Press, Boca Raton, 2008. (EN)
Plesník J., Dupačová J., Vlach M., Lineárne programovanie, Alfa, Bratislava, 1990. (CS)
Schmidtmayer J.: Maticový počet a jeho použití, SNTL, Praha, 1967. (CS)

eLearning

Classification of course in study plans

• Programme MPC-EEN Master's, any year of study, summer semester, elective

• Programme IT-MGR-2 Master's

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

• Programme MITAI Master's

specialization NADE , any year of study, summer semester, elective
specialization NBIO , any year of study, summer semester, elective
specialization NGRI , any year of study, summer semester, elective
specialization NNET , any year of study, summer semester, elective
specialization NVIZ , any year of study, summer semester, elective
specialization NCPS , any year of study, summer semester, elective
specialization NSEC , any year of study, summer semester, elective
specialization NEMB , any year of study, summer semester, elective
specialization NISD , any year of study, summer semester, elective
specialization NIDE , any year of study, summer semester, elective
specialization NISY , any year of study, summer semester, elective
specialization NMAL , any year of study, summer semester, elective
specialization NMAT , any year of study, summer semester, elective
specialization NSEN , any year of study, summer semester, elective
specialization NVER , any year of study, summer semester, elective
specialization NSPE , any year of study, summer semester, elective

• Programme MPC-AUD Master's

specialization AUDM-TECH , 1. year of study, summer semester, compulsory-optional
specialization AUDM-ZVUK , 1. year of study, summer semester, compulsory-optional

• Programme MPC-BIO Master's, 1. year of study, summer semester, compulsory-optional
• Programme MPC-TIT Master's, 1. year of study, summer semester, compulsory-optional
• Programme MPC-EVM Master's, 1. year of study, summer semester, compulsory-optional
• Programme MPC-EKT Master's, 1. year of study, summer semester, compulsory-optional
• Programme MPC-IBE Master's, 1. year of study, summer semester, compulsory

• Programme MITAI Master's

specialization NHPC , 1. year of study, summer semester, compulsory

• Programme MPC-MEL Master's, 1. year of study, summer semester, compulsory-optional
• Programme MPC-SVE Master's, 1. year of study, summer semester, compulsory-optional

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Computer-assisted exercise

18 hours, compulsory

Teacher / Lecturer

Project

8 hours, optionally

Teacher / Lecturer

eLearning