# Linear algerba II 2018

## Office hours

Office hours: by appointment (prefferably Tuesday or Wednesday). S322 (Malostranské nám. 25, 3rd floor).

## Lectures

Feb 22
Definition of norm (Section 7.2 of Poole) and inner product for real values (Section 7.1 of Poole) and complex values. Examples and main properties (Theorem 7.1 of Poole), Definition of length, distance and orhogonality. Pythagoras theorem (Theorem 7.2 of Poole), Orthogonal projection, Cauchy–Schwarz inequality.
Mar 2
Norm induced by the inner product satisfies triangle inequality (Theorem 7.4 of Poole), parallelogram law. Definition of an orthogonal and orthonormal set/base (Section 5.1 of poole but for inner products), Fourier coefficients (example in Poole). Gram–Schmidt orthogonalization (Poole sections 5.3 and 7.1).
Mar 9
Brief intro to Fourier transform, discrete Fourier transform and discrete cosine transform and applications (will not be on exam). Corollaries of Gram–Schmidt orthogonalization: every finitely generated inner produce space has orthonormal base; every set of orthonormal vectors extends to orthonormal base. Orthogonal complement in inner product space (Teorem 5.2 of poole for inner products rather than dot products). Properties of complement of sets and of complements of subspace (Thm 5.9 of Poole for inner produce spaces). Orthogonal projection: definition and orthogonal projection theorem.
Mar 16
Fundamental spaces of matrix: Row(A) is orthogonal to Null(A) (Thm 5.10 of Poole), corollary about fundamental spaces of ATT. Least square approximation (Section 7.3 of Poole). Review of linear transformations: Kernel and Range (Section 6.3 of Poole). Matrix of linear transormation. Composition of linear transformations is a matrix product. Inverse of linear transformation is inverse of a matrix. Definition of orthogonal and unitary matrix. Characterisaiton, examples.
Mar 23
This class will be replaced at Tuesday May 22.
Mar 30
Orthogonal matrices and linear transformations. Determinants - definition and basic properties. Effect of row operations on determinant. Computing determinant using REF. Matrix is regular iff its determinent is non-zero. Laplace method.
April 5
Cramer rule. Adjoint matirx. Computing inverses by determinnat. Geometric interpretation of determinants (volume and orientation)

Eigenvalues, eigenvectors - definition, geometric interpretation. Characterisation of eigenvalues. Characteristic polynomial. Eigenvalues are roots of the characteristic polynomial. Geometric and algebraic multiplicity of eigenvalue. Produc and sum of eigenvalues. Spectrum and dominating eigenvalue.

April 12
Properties of eigenvalues. Companion matrix (not needed for exam): finding eigenvalues of n x n matrix is as hard as finding roots of polynomials of degree n. Similar metrices. Diagonalization. Matrix is diagonalizable iff it has n linearly independent eigenvectors. Eigenvalues of AB and BA are the same. Examples: power of matrices, recurences.
April 19
Jordan normal form and properties: number of blocks in jordan form is number of eigenvectors (without proof). Corollary: algebraic muliplicity of eigenvalue is greater or equal the geometric multiplicity. Harmitian tranposition and hermitian matrix. Eigenvalues of symmetric real (and complex hermitian) are real. Spectral decomposition (orthogonal diagonalization)
April 27
Courant-Fisher min-max theorem as an application of spectral decomposition (not necessary for exam). Gershgorin circle theorem. Power method and its convergence (Section 4.5 of Poole's book). Google pagerank.
May 3
More on google pagerank (will not be part of an eam). Deflation of eigenvalues. Symmetric positive-semidefinite (PSD) and positive-definite matrices (PSD). Characterisation of positive-definite matrices. Properties of positive-definite matrices. Characterisation of positive semi-definite matrices.
May 10
Recurrence for testing postivie-definiteness. Cholesky decomposition. Algorithm for cholesky decomposition. Sylvester's criterrion for PSD. The correspondence between SPD and inner product.
Plan
Square-root of a matrix. Sylverster's critterion for PSD. Bilinear and quadratic forms: definition, matrix form, change of basis. Sylvester's law of inertia. Diagonalization of quadratic forms. Matrix decompositions.
Sample exam