Course detail
Matrices and Tensors Calculus
FEKT-MPC-MATAcad. year: 2025/2026
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
Number of ECTS credits
Mode of study
Guarantor
Department
Entry knowledge
Rules for evaluation and completion of the course
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
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
Prerequisites and corequisites
Basic literature
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)
Recommended reading
Davis, H. T., Thomson K. T., Linear Algebra and Linear Operators in Engineering, Academic Press, San Diego, ISBN 978-0122063497, 2007. (EN)
Demlová, M., Nagy, J., Algebra, STNL, Praha 1982. (CS)
Havel, V., Holenda J.: Lineární algebra, SNTL, Praha 1984. (CS)
Classification of course in study plans
- Programme MPC-NCP Master's 1 year of study, summer semester, compulsory-optional
- 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-EAK Master's 0 year of study, summer semester, elective
- Programme MPC-EEN Master's 0 year of study, summer semester, elective
- Programme MPC-EKT 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-IBE Master's 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
- Programme MPC-TIT Master's 1 year of study, summer semester, compulsory-optional
- Programme MITAI Master's
specialization NSEC , 0 year of study, summer semester, elective
specialization NISY up to 2020/21 , 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 NCPS , 0 year of study, summer semester, elective
specialization NHPC , 1 year of study, summer semester, compulsory
specialization NVER , 0 year of study, summer semester, elective
specialization NIDE , 0 year of study, summer semester, elective
specialization NISY , 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 NEMB , 0 year of study, summer semester, elective
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 NADE , 0 year of study, summer semester, elective
specialization NISD , 0 year of study, summer semester, elective
specialization NMAT , 0 year of study, summer semester, elective
Type of course unit
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Computer-assisted exercise
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