Course details
Matrices and Tensors Calculus
MPC-MAT FEKT MPC-MAT Acad. year 2024/2025 Summer semester 5 credits
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.
Guarantor
Language of instruction
Completion
Time span
- 26 hrs lectures
- 18 hrs exercises
- 8 hrs projects
Department
Learning objectives
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
Prerequisite knowledge and skills
The knowledge of the content of the subject Matematika 1 is required. The previous attendance to the subject Matematický seminář is warmly recommended.
Study literature
- Crandal, R. E., Mathematica for the Sciences, Addison-Wesley, Redwood City, ISBN 978-0201510010, 1991.
- Davis, H. T., Thomson K. T., Linear Algebra and Linear Operators in Engineering, Academic Press, San Diego, ISBN 978-0122063497, 2007.
- Mannuci, M. A., Yanofsky N. S., Quantum Computing For Computer Scientists, Cambridge University Press, Cabridge, ISBN 978-0521879965, 2008.
- Nahara, M., Ohmi T., Quantum Computing: From Linear Algebra to Physical Realizations, CRC Press, Boca Raton, ISBN 978-0750309837, 2008.
Fundamental literature
- Kolman, B., Elementary Linear Algebra, Macmillan Publ. Comp., New York, ISBN 978-0029463703, 1986.
- Kolman, B., Hill, D. R., Introductory Linear Algebra, Pearson, New York, 978-8131723227, 2008.
- Kovár, M., Selected Topics on Multilinear Algebra with Applications, Skriptum, Brno, 2015, 141s.
Syllabus of lectures
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.
Syllabus of numerical exercises
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.
Syllabus - others, projects and individual work of students
Dva projekty na vybraná témata z aplikované matematiky, každý po 5 bodech.
Progress assessment
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.
Course inclusion in study plans