Course detail

Matrices and Tensors Calculus

FEKT-MPC-MATAcad. year: 2026/2027

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. Matice jako algebraická struktura. Operace s maticemi. Determinant. Matice v soustavách lineárních algebraických rovnic. Vektorový prostor, báze a dimenze. Transformace souřadnic. Součet a průnik vektorových prostorů. Lineární zobrazení vektorových prostorů a jeho maticové vyjádření. Skalární součin, ortogonální průmět a prvek nejlepší aproximace. Problém vlastních hodnot. Spektrální vlastnosti (zejména samoadjungovaných) matic. Bilineární a kvadratické formy, definitnost kvadratických forem. Lineární formy a tenzory. Různé typy souřadnic. Kovariantní, kontravariantní a smíšené tenzory. Operace s tenzory. Tenzorový antisymetrický vnější součin. Antilineární formy. Lorentzova transformace jako důsledek spektrálních vlastností matic. Maticová formulace kvantové mechaniky. Diracova notace. Bra a Ket vektory. Vlnové pakety jako vektory. Samoadjungovaný lineární operátor. Schrodingerova rovnice. Princip neurčitosti a Heisenbergova relace. Multi-qubitové systémy a kvantová provázanost (entaglement). Einstein-Podolsky-Rosen experiment-paradox. Kvantové výpočty. Matice hustoty. Kvantová teleportace.

Language of instruction

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

doc. RNDr. Martin Kovár, Ph.D.

Department

Department of Mathematics (UMAT)

Entry knowledge

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

Rules for evaluation and completion of the course

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.
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.

Aims

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

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Kolman, B., Elementary Linear Algebra, Macmillan Publ. Comp., New York, ISBN 978-0029463703, 1986. (EN)
Kolman, B., Hill, D. R., Introductory Linear Algebra, Pearson, New York, 978-8131723227, 2008. (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)

Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

doc. RNDr. Martin Kovár, Ph.D.

Syllabus

Matice a maticové operace. Determinat. Soustavy lineárních rovnic.
Vektorové prostory, báze, dimenze transformace souřadnic.
Operace s vektorovými prostory - součet, průnik a lineární zobrazení.
Skalární součin, ortogonální průmět a prvek nejlepší aproximace.
Problém vlastních hodnot. Spektrální vlastnosti matic.
Bilineární a kvadratické formy, definitnost kvadratických forem.
Lineární formy a tenzory. Různé typy tenzorů a souřadnic.
Operace s tenzory - tenzorový a vnější součin.
Fyzikální aplikace - Lorentzova transformace, maticová kvantová mechanika.

Computer-assisted exercise

26 hours, compulsory

Teacher / Lecturer

doc. RNDr. Martin Kovár, Ph.D.

Syllabus