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

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

Prerequisites

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.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of 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

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.

Not applicable.

Classification of course in study plans

• Programme AUDIO-P Master's

branch P-AUD , 2 year of study, summer semester, elective interdisciplinary
branch P-AUD , 1 year of study, summer semester, elective interdisciplinary

• Programme IBEP-V Master's

branch V-IBP , 1 year of study, summer semester, compulsory

• Programme EEKR-M Master's

branch M-EEN , 1 year of study, summer semester, theoretical subject
branch M-KAM , 1 year of study, summer semester, theoretical subject
branch M-TIT , 1 year of study, summer semester, theoretical subject
branch M-SVE , 1 year of study, summer semester, theoretical subject
branch M-EST , 1 year of study, summer semester, theoretical subject
branch M-EVM , 1 year of study, summer semester, theoretical subject

• Programme IT-MSC-2 Master's

branch MBI , 0 year of study, summer semester, elective
branch MBS , 0 year of study, summer semester, elective
branch MGM , 0 year of study, summer semester, elective
branch MIN , 0 year of study, summer semester, elective
branch MIS , 0 year of study, summer semester, elective
branch MMI , 0 year of study, summer semester, elective
branch MMM , 0 year of study, summer semester, elective
branch MPV , 0 year of study, summer semester, elective
branch MSK , 0 year of study, summer semester, elective

• Programme MITAI Master's

specialization NHPC , 1 year of study, summer semester, compulsory
specialization NBIO , 0 year of study, summer semester, elective
specialization NSEN , 0 year of study, summer semester, elective
specialization NVIZ , 0 year of study, summer semester, elective
specialization NGRI , 0 year of study, summer semester, elective
specialization NISD , 0 year of study, summer semester, elective
specialization NSEC , 0 year of study, summer semester, elective
specialization NCPS , 0 year of study, summer semester, elective
specialization NNET , 0 year of study, summer semester, elective
specialization NMAL , 0 year of study, summer semester, elective
specialization NVER , 0 year of study, summer semester, elective
specialization NIDE , 0 year of study, summer semester, elective
specialization NEMB , 0 year of study, summer semester, elective
specialization NSPE , 0 year of study, summer semester, elective
specialization NADE , 0 year of study, summer semester, elective
specialization NMAT , 0 year of study, summer semester, elective
specialization NISY , 0 year of study, summer semester, elective

• Programme EEKR-CZV lifelong learning

branch EE-FLE , 1 year of study, summer semester, theoretical subject

#### Type of course unit

Lecture

26 hod., optionally

Teacher / Lecturer

Syllabus

Definition of matrix, fundamental notion. Transposition of matrices.
Operations with matrices. Special types of matrices. Inverse matrix.
Matrix solutions of linear algebraic equations.
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.

Computer-assisted exercise

18 hod., compulsory

Teacher / Lecturer

Syllabus

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