Linear Algebra 2 - NMAI058
Title: Lineární algebra 2
Guaranteed by: Department of Applied Mathematics (32-KAM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019
Semester: summer
E-Credits: 5
Hours per week, examination: summer s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Guarantor: doc. Mgr. Milan Hladík, Ph.D.
doc. Mgr. Petr Kolman, Ph.D.
Class: Informatika Bc.
Classification: Mathematics > Algebra
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Annotation -
Last update: FIALA/MFF.CUNI.CZ (17.02.2010)
Continuation of MAI057 - special matrices, determinants, eigenvalues, examples of applications of linear algebra.
Course completion requirements -
Last update: Mgr. Jan Hubička, Ph.D. (28.02.2018)

For passing the class you should obtain at least 70% of total marks from the tests given during the semester and a practical programming homework. Students can obtain addidional marks by homework (the amount of homework depends on missing marks). Homework will be given at the end of the semester if someone needs it.

Literature -
Last update: doc. Mgr. Milan Hladík, Ph.D. (22.11.2012)

W. Gareth. Linear Algebra with Applications. Jones and Bartlett Publishers, Boston, 4th edition, 2001.

C. D. Meyer. Matrix analysis and applied linear algebra. SIAM, Philadelphia, PA, 2000.

G. Strang. Linear algebra and its applications. Thomson, USA, 4rd edition, 2006.

Requirements to the exam -
Last update: Mgr. Jan Hubička, Ph.D. (28.02.2018)

There will an examination consisting of a written and oral part.

The written part involves several exercises. The oral part involves discussion of solutions to the set problems and additional questions on topics covered in lectures and classes.

Class credits ("zapocet") are prerequisite for taking the examination

Syllabus -
Last update: doc. Mgr. Milan Hladík, Ph.D. (11.05.2020)

Inner product spaces:

  • norm induced by an inner product
  • Pythagoras theorem, Cauchy-Schwarz inequality, triangle inequality
  • orthogonal and orthonormal system of vectors, Fourier coefficients, Gram-Schmidt orthogonalization
  • orthogonal complement, orthogonal projection
  • the least squares method
  • orthogonal matrices


  • basic properties
  • Laplace expansion of a determinant, Cramer's rule
  • adjugate matrix
  • geometric interpretation of determinants

Eigenvalues and eigenvectors:

  • basic properties, characteristic polynomial, companion matrix
  • Cayley-Hamilton theorem
  • similarity and diagonalization of matrices, spectral decomposition, Jordan normal form
  • symmetric matrices and their spectral decomposition
  • estimation and computation of eigenvalues: Gershgorin discs and power method

Positive semidefinite and positive definite matrices:

  • characterization and properties
  • methods: recurrence formula, Cholesky decomposition, Gaussian elimination, Sylvester's criterion
  • relation to inner products

Bilinear and quadratic forms:

  • forms and their matrices, change of a basis
  • Sylvester's law of inertia, diagonalization, polar basis

Topics on expansion:

  • eigenvalues of nonnegative matrices
  • matrix decompositions: Householder transformation, QR, SVD, Moore-Penrose pseudoinverse of a matrix