Home People Schenato Teaching Applied Linear Algebra

 


APPLIED LINEAR ALGEBRA

a.a. 2020-2021

Ph.D. School in Information Engineering


 

tl_files/utenti/lucaschenato/Figure/square.png Instructor

Prof. Luca Schenato
Phone: 049 827 7925
E-mail:
 ( NO luca.schenato@dei.unipd.it !!!!)
Webpage: http://automatica.dei.unipd.it/people/schenato.html
 

tl_files/utenti/lucaschenato/Figure/square.png Description
  • Vectors: inner products, norms, main operations (average, standard deviation, ...)
  • Matrices: matrix-vector and matrix-matrix multiplication, Frobenius norm,
  • Complexity, sparsity
  • Special matrices: Diagonal, Upper Triangular, Lower triangular, Permutation (general pair), inverse and orthogonal
  • A square and invertible: LU decomposition (aka gaussian elimination), LU-P decomposition, Cholesky decomposition
  • Ax=b via LU-P decomposition: forward and backward substitution
  • (sub)Vector spaces: definitions, span, bases (standard, orthogonal, orthonormal), dimension, direct sum, orthogonal complement, null space, orthogonal complement theorem
  • Gram-Smith orthogonalization and QR decomposition (square and invertible A, general non-square)
  • Ax=b via QR decomposition. LU-P vs QR
  • Linear maps: image space, kernel, column and row rank
  • Fundamental Theorem of Linear Algebra (Part I): rank-nullity Theorem, the 4 fundamental subspace
  • Eigenvalues/eigenvector and Shur decomposition
  • Projection matrices: oblique and orthogonal, properties
  • Positive semidefinite matrices: properties and quadratic functions square root matrix
  • Properties of A'A and AA' and Polar decomposition
  • Singular Value Decomposition: proofs and properties
  • Pseudo-inverse: definition and relation to SVD
  • Fundamental Theorem of Linear Algebra (Part II): special orthogonal basis for diagonalization
  • Least-Squares: definition, solution and algorithms
  • Ill-conditioned problems vs stability of algorithms, numerical conditioning

 

tl_files/utenti/lucaschenato/Figure/square.png Objectives
  • Theory: formal proofs of many results (theorem-proof type problems)
  • Algorithms: understanding of most commonly used algortithm used in MATLAB and Python for Linear Algebra
  • Implementation: MATLAB implementation of algorithms and performance evaluation on Big Data
tl_files/utenti/lucaschenato/Figure/square.png Lectures

 

Week

TUESDAY

(10:30-12:30)

FRIDAY

(10:30-12:30)

1 (17-19/11)

Course introduction. Vectors and matrices (Lecture 1)

LU decomposition and solution of square systems (Lecture 2)

2 (24-26/11)

Vector Subspaces (Lecture 3)

QR orthogonalization and decomposition (Lecture 4)

3 (1-3/12)

Linear maps and Fundamental Theorem of Linear Algebra (Lecture 5) Eigenvectors, Shur Decomposition, Projections (Lecture 6)

4 (8-10/12)

NO CLASS
Positive Definite matrices, roots and quadratic function (Lecture 7) 

5 (15-17/12)

Polar Decomposition and SVD (Lecture 8)  Pseudo-Inverse and Least squares (Lecture 9) N
6 (22/12

Numerical conditioning (Lecture 10)

 

 

tl_files/utenti/lucaschenato/Figure/square.png References

Textbooks and Internet Notes:

  1. S. Boyd, L. Vanderberghe, "Introduction to Applied Linear Algebra", Cambridge University Press, 2018
  2. G. Strang, "The Fundamental Theorem of Linear Algebra", The American Mathematical Monthly, vol. 100(9), pp. 848-855, 1993
  3. G. Strang, "Linear Algebra and Learning From Data", Wellesley - Cambridge Press, 2019

 

tl_files/utenti/lucaschenato/Figure/square.png Final Exam Grading
  1. Homeworks
  2. Written final exam
  3. Short presentation based on a recent paper of Linear Algebra Algorithms for Big Data