Advanced Linear Algebra - Algorithms
Niall Madden School of Mathematics, Statistics and Applied Mathematics National University of Ireland, GalwayNiall.Madden@NUIGalway.ie
November 27, 2017
These notes accompany the Algorithms section of Advanced Linear Algebra (MA500-1) taught at NUI Galway in Semester 1 2017/2018 by Niall Madden. For notes of the theory section, taught by Rachel Quinlan, please see http://www.maths.nuigalway.ie/~rquinlan/linearalgebra/ .
1 Introduction (RQ) 2 Matrix Multiplication (NM) 3 Range, nullspace, and rank (NM) 4 The matrix of a linear transformation (RQ) 5 Some bases are better than others (RQ) 6 Orthogonal Vectors (NM) 7 Unitary Matrices (NM, 18 Sep) 8 Symmetric Matrices I (RQ, 19 Sep) 9 Symmetric Matrices II (RQ, 20 Sep) 10 Vector Norms (NM, 25 Sep) 11 Matrix Norms (NM, 26 Sep) 12 Diagonalizability of real symmetric matrices (RQ, 27 Sep) 13 Introduction to the Singular Value Decomposition (NM, 2 Oct) 14 No class (3 Oct) 15 Tutorial (RQ, 4 Oct) 16 Positive (semi)definite matrices (RQ, 9 Oct) 17 Positive (semi)definite matrices (RQ, 10 Oct) 18 Existence of the Singular Value Decomposition (NM, 11 Oct) 19 Ophelia (N/A, 16 Oct) 20 Yet more about the SVD (NM, 17 Oct) 21 Stuff (RQ, 18 Oct) 22 Some theorems about the SVD (NM, 23 Oct) 23 Low Rank Approximations (NM, 24 Oct) 24 Positive matrices - the Perron-Frobenius Theorem. (RQ, 25 Oct) 25 Proof of the Perron-Frobenius Theorem. (RQ, 31 Oct) 26 Projectors (NM, 1 Nov) 27 Orthogonal Projectors (NM, 6 Nov) 28 More on the proof of the Perron-Frobenius Theorem. (RQ, 7 Nov) 29 The end of the proof of the Perron-Frobenius Theorem. (RQ, 8 Nov) 30 The \(QR\) factorisation (NM, 13 Nov) 31 The Schur Form (NM, 14 Nov) 32 Applications of Perron-Frobenius (I). (RQ, 15 Nov) 33 Applications of Perron-Frobenius (II). (RQ, 20 Nov) 34 Hessenberg Form and the QR Algorithm. (NM, 21 Nov) 35 Review (RQ + NM, 22 Nov) References