Course detail

Matrices and tensors calculus

FEKT-NMATAcad. year: 2018/2019

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

English

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. RNDr. Martin Kovár, Ph.D.

Department

Department of Mathematics (UMAT)

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

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

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.
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.

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

Kolman, B., Elementary Linear Algebra, Macmillan Publ. Comp., New York 1986.
Kolman, B., Introductory Linear Algebra, Macmillan Publ. Comp., New York 1991.

Recommended reading

Davis H. T., Thomson K. T., Linear Algebra and Linear Operators in Engineering, Academic Press, San Diego, 2007.
Gantmacher, F. R., The Theory of Matrices, Chelsea Publ. Comp., New York 1960.

Classification of course in study plans

  • Programme AUDIO-PU Master's

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

  • Programme EECC-MN Master's

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

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Martin Kovár, Ph.D.

Syllabus

Definition of matrix, fundamental notion. Transposition of matrices.
Determinant of quadratic complex matrix.
Operations with matrices. Special types of matrices. Inverse matrix.
Matrix solutions of linear algebraic equations.
Linear, bilinear and quadratic forms. Definite of quadratics forms.
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

doc. RNDr. Martin Kovár, Ph.D.

Syllabus

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