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APPLIED LINEAR ALGEBRA
a.a. 2020-2021 Ph.D. School in Information Engineering |
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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
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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
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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
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Lectures |
| Week |
TUESDAY (10:30-12:30) |
FRIDAY (10:30-12:30) |
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1 (17-19/11) |
Course introduction. Vectors and matrices (Lecture 1) |
LU decomposition and solution of square systems (Lecture 2) |
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2 (24-26/11) |
Vector Subspaces (Lecture 3) |
QR orthogonalization and decomposition (Lecture 4) |
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3 (1-3/12) |
Linear maps and Fundamental Theorem of Linear Algebra (Lecture 5) | Eigenvectors, Shur Decomposition, Projections (Lecture 6) |
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4 (8-10/12) |
NO CLASS |
Positive Definite matrices, roots and quadratic function (Lecture 7) |
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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) |
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References |
Textbooks and Internet Notes:
- S. Boyd, L. Vanderberghe, "Introduction to Applied Linear Algebra", Cambridge University Press, 2018
- G. Strang, "The Fundamental Theorem of Linear Algebra", The American Mathematical Monthly, vol. 100(9), pp. 848-855, 1993
- G. Strang, "Linear Algebra and Learning From Data", Wellesley - Cambridge Press, 2019
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Final Exam Grading |
- Homeworks
- Written final exam
- Short presentation based on a recent paper of Linear Algebra Algorithms for Big Data