results for different matrices and comparisons dense
play

Results for different matrices and comparisons Dense Matrices - PowerPoint PPT Presentation

Mar 07, 2024 •251 likes •515 views

Results for different matrices and comparisons Dense Matrices Rectangular Matrices Rectangular Matrices Symmetric Matrices Sparse Matrices Dense Rectangular matrices Dense Rectangular matrices Dense Rectangular matrices Dense

  1. Results for different matrices and comparisons

  2. • Dense Matrices – Rectangular Matrices Rectangular Matrices – Symmetric Matrices • Sparse Matrices

  3. Dense Rectangular matrices Dense Rectangular matrices

  4. Dense Rectangular matrices Dense Rectangular matrices • Comparison of timings when m is fixed and n is varied n m Built in Comb. Comb. Indirect Direct Francis Built in Square Method Method SVD 100 0.0073 0.1779 0.5651 0.4488 0.1616 0.8897 200 0.0129 0.4382 1.6292 1.1313 0.4599 1.3890 700 700 0 0451 0.0451 11 4075 11.4075 25.4917 25 4917 14 3346 14.3346 11 6559 11.6559 3 6083 3.6083 100 800 0.0589 16.5906 35.1912 19.3266 16.7657 3.9636 1400 1.1683 76.9117 147.1903 71.8148 75.7396 7.6926 1500 1.3396 96.2016 181.0894 93.1726 94.5906 8.2684 100 0.0191 0.4518 1.6380 1.1398 0.4642 1.8997 200 0.0331 1.7899 3.5367 1.8215 1.9187 5.5780 700 0.1123 26.2992 40.5601 14.7202 26.1704 14.6077 200 800 0.1689 37.0253 55.9529 19.7481 37.0049 16.2683 1400 0.2952 162.2110 236.5831 74.3360 162.2961 27.0306 1500 0.3177 200.9886 290.3429 89.6555 201.3425 28.6722

  5. Dense Rectangular matrices Dense Rectangular matrices

  6. Dense Rectangular matrices Dense Rectangular matrices • Comparison of accuracies when n is fixed and m is varied Comb. Comb. Indirect Direct n m Built in Built in Square Method Method SVD 2.22e ‐ 13 9e ‐ 13 9.3e ‐ 13 5.5e ‐ 13 100 3.8e-13 200 5.66e ‐ 13 8.1e-13 8e ‐ 13 3e ‐ 13 13.7e ‐ 13 700 20.88e ‐ 13 30.04e ‐ 13 28e ‐ 13 10.2e ‐ 13 36.5e ‐ 13 100 25.78e ‐ 13 37.6e ‐ 13 33e ‐ 13 12.1e ‐ 13 46.7e ‐ 13 800 40.33e ‐ 13 67.6e ‐ 13 61e ‐ 13 25.1e ‐ 13 82.3e ‐ 13 1400 1500 47.24e ‐ 13 70.9e ‐ 13 60e ‐ 13 27.6e ‐ 13 84.1e ‐ 13 100 3.54e ‐ 13 5.6e ‐ 13 5e ‐ 13 1.6e ‐ 13 7.3e ‐ 13 5.69e 13 5 69e ‐ 13 11 3e ‐ 13 11.3e 13 1482e 13 1482e ‐ 13 247 6e ‐ 13 247.6e 13 18e ‐ 13 18e 13 200 200 20.56e ‐ 13 43.9e ‐ 13 43e ‐ 13 12.1e ‐ 13 52.6e ‐ 13 700 200 800 25.42e ‐ 13 50.8e ‐ 13 51e ‐ 13 14.3e ‐ 13 59.8e ‐ 13 1400 44.68e ‐ 13 98.3e ‐ 13 98e ‐ 13 27.6e ‐ 13 115.5e ‐ 13 48.22e ‐ 13 107.3e ‐ 13 102e ‐ 13 30.1e ‐ 13 129.3e ‐ 13 1500

  7. Dense Rectangular matrices Dense Rectangular matrices

  8. Dense Rectangular matrices Dense Rectangular matrices • Comparison of accuracies when m is fixed and n is varied n m Built in Comb. Comb. Indirect Direct Built in Square Method Method SVD 2.62 e-13 3.7 e-13 8.2 e-13 5.92 e-13 5.8 e-13 100 3.04 e-13 5.26 e-13 4.8 e-13 1.55 e-13 6.04 e-13 200 700 700 2 93 13 2.93 e-13 8.15 e-13 8 1 13 8.2 e-13 8 2 13 1 8 13 1.78 e-13 7.87 e-13 8 13 100 800 3.97 e-13 9.8 e-13 9.6 e-13 1.99 e-13 7.45 e-13 3.52 e-13 12.34 e-13 12 e-13 2.39 e-13 8.97 e-13 1400 3.2 e-13 17.57 e-13 17.1 e-13 2.31 e-13 10.59 e-13 1500 100 6.04 e-13 9.21 e-13 8.1 e-13 3.17 e-13 14.07 e-13 200 5.28 e-13 11.53 e-13 247.7 e-13 16.71 e-13 17.68 e-13 6.51 e-13 18.67 e-13 18.4 e-13 4.05 e-13 20.09 e-13 700 200 6.20 e-13 19.55 e-13 19.4 e-13 4.03 e-13 18.89 e-13 800 1400 5.86 e-13 21.5 e-13 21.5 e-13 4.38 e-13 21.15 e-13 1500 6.17 e-13 27.02 e-13 26.5 e-13 4.9 e-13 24.03 e-13

  9. Dense Rectangular matrices Dense Rectangular matrices

  10. Dense Rectangular matrices Dense Rectangular matrices

  11. Dense Rectangular matrices Dense Rectangular matrices

  12. Dense Rectangular matrices Dense Rectangular matrices • Orthogonality checks when m is fixed and n is varied m n Built in SVD Comb. Built in Comb. Square Indirect Method Direct Method UU T ‐ Id VV T ‐ Id UU T ‐ Id VV T ‐ Id UU T ‐ Id VV T ‐ Id UU T ‐ Id VV T ‐ Id UU T ‐ Id VV T ‐ Id 0.28 e-13 0.22 e-13 0.03 e-13 0.77 e-13 0.71 e-13 100 0.19e-13 0.18e-13 0.28e-13 0.22e-13 0.03e-13 200 0.22 e-13 0.18 e-13 0.71 e-13 0.33 e-13 0.49 e-13 0.24 e-13 0.07 e-13 0.04 e-13 0.71 e-13 0.42 e-13 700 0.37 e-13 0.20 e-13 1.45 e-13 0.31 e-13 1.05 e-13 0.23 e-13 0.17 e-13 0.03 e-13 1.44 e-13 0.47 e-13 100 1.59 e-13 1.05 e-13 0.03 e-13 1.59 e-13 0.34 e-13 800 0.30 e-13 0.19 e-13 0.28 e-13 0.21 e-13 0.19 e-13 1 91 13 1.91 e-13 1.56 e-13 1 56 13 0 03 13 0.03 e-13 1.91 e-13 1 91 13 0.46 e-13 0 46 13 1400 1400 0.53 e-13 0 53 13 0 21 13 0.21 e-13 0 30 13 0.30 e-13 0.21 e-13 0 21 13 0.33 e-13 0 33 13 1500 0.51 e-13 0.19 e-13 1.79 e-13 0.31 e-13 1.82 e-13 0.22 e-13 0.35 e-13 0.03 e-13 1.79 e-13 0.37 e-13 100 0.20e-13 0.21e-13 0.31 e-13 0.70e-13 0.22 e-13 0.48e-13 0.04e-13 0.06 e-13 0.37 e-13 0.70 e-13 0.62 e-13 0.53 e-13 0.07 e-13 4.66 e-13 4.70 e-13 200 0.32 e-13 0.30 e-13 0.66 e-13 0.55 e-13 0.08 e-13 2.72 e-13 1.56 e-13 0.07 e-13 2.73 e-13 0.74 e-13 700 0.40 e-13 0.31 e-13 0.64 e-13 0.54 e-13 0.18 e-13 200 800 0.43 e-13 0.30 e-13 2.92 e-13 0.67 e-13 1.47 e-13 0.56 e-13 0.20 e-13 0.07 e-13 2.92 e-13 0.76 e-13 1400 0.57 e-13 0.30 e-13 3.06 e-13 0.74 e-13 1.97 e-13 0.76 e-13 0.32 e-13 0.06 e-13 3.05 e-13 0.81 e-13 3.56 e-13 2.16 e-13 0.06 e-13 3.55 e-13 0.80 e-13 1500 0.61 e-13 0.29 e-13 0.66 e-13 0.56 e-13 0.34 e-13

  13. Dense Rectangular matrices Dense Rectangular matrices

  14. Dense Rectangular matrices Dense Rectangular matrices

  15. Dense Rectangular matrices Dense Rectangular matrices • Error in singular values when m is fixed and n is varied n m Comb. Comb. Indirect Direct Francis Built in Square Method Method 0.42e ‐ 13 2.96e ‐ 13 2.42e ‐ 13 0.92e ‐ 13 2.59e ‐ 13 100 0.56e ‐ 13 0.37e ‐ 13 0.24e ‐ 13 0.56e ‐ 13 1.53e ‐ 13 200 700 700 0 92 0.92e ‐ 13 13 0 64 0.64e ‐ 13 13 0.71e ‐ 13 0 1 13 1.06e ‐ 13 1 06 13 2 49 2.49e ‐ 13 13 100 800 0.71e ‐ 13 0.60e ‐ 13 0.67e ‐ 13 1.56e ‐ 13 2.42e ‐ 13 0.99e ‐ 13 1.78e ‐ 13 1.20e ‐ 13 1.49e ‐ 13 2.98e ‐ 13 1400 0.71e ‐ 13 0.85e ‐ 13 0.63e ‐ 13 1.49e ‐ 13 5.76e ‐ 13 1500 100 0.67e ‐ 13 0.43e ‐ 13 0.32e ‐ 13 0.74e ‐ 13 2.63e ‐ 13 200 0.56e ‐ 13 87e ‐ 13 3.34e ‐ 13 1.24e ‐ 13 5.51e ‐ 13 0.78e ‐ 13 0.71e ‐ 13 0.71e ‐ 13 1.70e ‐ 13 8.6e ‐ 13 700 200 0.99e ‐ 13 0.78e ‐ 13 0.56e ‐ 13 1.45e ‐ 13 11.65e ‐ 13 800 1400 0.85e ‐ 13 0.92e ‐ 13 0.92e ‐ 13 2.06e ‐ 13 9.73e ‐ 13 1500 0.71e ‐ 13 1.28e ‐ 13 1.13e ‐ 13 1.92e ‐ 13 15.49e ‐ 13

  16. Dense Symmetric matrices Dense Symmetric matrices • Comparison of timings Comparison of timings Comb. Built in Comb. Square Indirect Direct Method Francis Built in SVD n Method 100 100 0.1478 0.1478 0.8 0.8 1.5 1.5 1.0154 1.0154 0.3 0.3 1.1 1.1 200 0.0396 3.3 6.9 4.3111 3 6.3 700 1.1737 359.7 414.4 57.9199 366.7 210.8 800 800 1.6642 66 2 603 603.4 6 675.6 6 77.6638 6638 622 2 622.2 3 3 313.4 1400 9.3720 5518.4 5761.6 73.7534 5655.7 1622.8 1500 11.2318 7256.0 7546.5 321.702 7405.2 1988.7

  17. Dense Symmetric matrices Dense Symmetric matrices • Comparison of Accuracies Comparison of Accuracies Built in SVD Comb. Built in Comb. Square Indirect Method Direct Method n 100 100 0.35e ‐ 12 0 35e ‐ 12 0 62e ‐ 12 0.62e ‐ 12 1.171e ‐ 12 1 171e ‐ 12 0 258e ‐ 12 0.258e ‐ 12 0 91e ‐ 12 0.91e ‐ 12 200 0.83e ‐ 12 1.83e ‐ 12 8.027e ‐ 12 1.11e ‐ 12 2.59e ‐ 12 700 3.85e ‐ 12 12.88e ‐ 12 84.79e ‐ 12 24.28e ‐ 12 16.96e ‐ 12 800 4.66e ‐ 12 16.25e ‐ 12 75.42e ‐ 12 40.11e ‐ 12 20.10e ‐ 12 1400 10.53e ‐ 12 40.89e ‐ 12 781.6e ‐ 12 910.0e ‐ 12 51.26e ‐ 12 1500 10.92e ‐ 12 44.37e ‐ 12 133.4e ‐ 12 126.2e ‐ 12 55.05e ‐ 12

  18. Dense Symmetric matrices Dense Symmetric matrices • Comparison of Orthogonality checks ||UU T ‐ Id|| Comparison of Orthogonality checks ||UU ‐ Id|| Built in SVD Comb. Built in Comb. Square Indirect Method Direct Method n 100 100 0.19e ‐ 13 0 19e ‐ 13 0 35e ‐ 13 0.35e ‐ 13 0.29e ‐ 13 0 29e ‐ 13 0 03e ‐ 13 0.03e ‐ 13 0 53e ‐ 13 0.53e ‐ 13 200 0.33e ‐ 13 0.69e ‐ 13 0.65e ‐ 13 0.06e ‐ 13 0.969e ‐ 13 700 0.89e ‐ 13 2.56e ‐ 13 2.52e ‐ 13 0.20e ‐ 13 3.40e ‐ 13 800 1.05e ‐ 13 2.87e ‐ 13 2.80e ‐ 13 0.22e ‐ 13 6.80e ‐ 13 1400 1.63e ‐ 13 5.94e ‐ 13 5.77e ‐ 13 0.36e ‐ 13 40.63e ‐ 13 1500 1.68e ‐ 13 7.04e ‐ 13 6.92e ‐ 13 0.38e ‐ 13 8.348e ‐ 13

  19. Dense Symmetric matrices Dense Symmetric matrices • Comparison of error in singular values Comparison of error in singular values Built in SVD Comb. Built in Comb. Square Indirect Method Direct Method n 100 100 0.03e ‐ 12 0 03e ‐ 12 0 421e ‐ 12 0.421e ‐ 12 0.29e ‐ 12 0 29e ‐ 12 0 12e ‐ 12 0.12e ‐ 12 0 13e ‐ 12 0.13e ‐ 12 200 0.07e ‐ 12 2.999e ‐ 12 0.65e ‐ 12 0.25e ‐ 12 0.78e ‐ 12 700 0.15e ‐ 12 27.92e ‐ 12 2.52e ‐ 12 0.49e ‐ 12 4.59e ‐ 12 800 0.14e ‐ 12 22.87e ‐ 12 2.80e ‐ 12 0.76e ‐ 12 8.27e ‐ 12 1400 0.18e ‐ 12 226.2e ‐ 12 5.77e ‐ 12 1.25e ‐ 12 12.32e ‐ 12 1500 0.22e ‐ 12 34.41e ‐ 12 6.92e ‐ 12 1.13e ‐ 12 21.03e ‐ 12

Recommend

Dense matrix algorithms We are going to study algorithms involving dense matrices (as opposed

Dense matrix algorithms We are going to study algorithms involving dense matrices (as opposed

Dense matrix algorithms We are going to study algorithms involving dense matrices (as opposed to sparse matrices ) A very important issue is how to map a matrix onto processors the combination of proper mapping and efficient algorithm

507 views • 8 slides
Sparse Matrices sparse many elements are zero dense few elements are zero Example Of

Sparse Matrices sparse many elements are zero dense few elements are zero Example Of

Sparse Matrices sparse many elements are zero dense few elements are zero Example Of Sparse Matrices diagonal tridiagonal lower triangular (?) These are structured sparse matrices. May be mapped into a 1D array so that a mapping

698 views • 11 slides
Compression, inversion and sparse approximate PCA of dense kernel matrices in near linear

Compression, inversion and sparse approximate PCA of dense kernel matrices in near linear

Compression, inversion and sparse approximate PCA of dense kernel matrices in near linear computational complexity Florian Schfer ICERM 2017 F. Schfer, T.J. Sullivan, H. Owhadi Sparse factorisation of dense Kernel matrices June 8th 2017

1.68k views • 130 slides
Solving Large Dense Linear Systems with Covariance Matrices Jie Chen Mathematics and Computer

Solving Large Dense Linear Systems with Covariance Matrices Jie Chen Mathematics and Computer

Solving Large Dense Linear Systems with Covariance Matrices Jie Chen Mathematics and Computer Science Division Argonne National Laboratory Solving Large Dense Linear Systems with Covariance Matrices p. 1/30 SIAM LA, June 19, 2012

553 views • 30 slides
Code-to-Code Comparisons for the Problem of Shock Acceleration of a Diffuse Dense Gaseous

Code-to-Code Comparisons for the Problem of Shock Acceleration of a Diffuse Dense Gaseous

Code-to-Code Comparisons for the Problem of Shock Acceleration of a Diffuse Dense Gaseous Cylinder J.A. Greenough 1 , W.J. Rider 2 , C. Zoldi 2 , J.R. Kamm 2 8 th IWPCTM December 9-14, 2001 Pasadena, CA 1 Lawrence Livermore National Laboratory,

477 views • 26 slides
Sparse Matrices Example Of Sparse Matrices diagonal tridiagonal sparse many elements are

Sparse Matrices Example Of Sparse Matrices diagonal tridiagonal sparse many elements are

Sparse Matrices Example Of Sparse Matrices diagonal tridiagonal sparse many elements are zero lower triangular (?) dense few elements are zero These are structured sparse matrices. May be mapped into a 1D array so that a mapping

391 views • 6 slides
Multiple Comparisons Occasionally, e.g., at the start of a research project, we do not have a

Multiple Comparisons Occasionally, e.g., at the start of a research project, we do not have a

Lecture 7.1: Multiple Comparisons (A non-quiz topic) Examples of the need for multiple comparisons The problem with multiple comparisons post hoc; an outline of a solution Specific solutions: Fisher LSD, Tukey HSD, Holm (same as

840 views • 69 slides
Lightweight MDS Involution Matrices Siang Meng Sim 1 Khoongming Khoo 2 erique Oggier 1 Thomas

Lightweight MDS Involution Matrices Siang Meng Sim 1 Khoongming Khoo 2 erique Oggier 1 Thomas

Introduction Analyzing XOR Count E.C. of Had-based Matrices Comparisons Lightweight MDS Involution Matrices Siang Meng Sim 1 Khoongming Khoo 2 erique Oggier 1 Thomas Peyrin 1 Fr ed 1.Nanyang Technological University, Singapore 2.DSO

877 views • 36 slides
MATHEMATICS 1 CONTENTS Matrices Special matrices Operations with matrices Matrix

MATHEMATICS 1 CONTENTS Matrices Special matrices Operations with matrices Matrix

Matrices BUSINESS MATHEMATICS 1 CONTENTS Matrices Special matrices Operations with matrices Matrix multipication More operations with matrices Matrix transposition Symmetric matrices Old exam question Further study 2 MATRICES A

615 views • 25 slides
Some results and problems on Countable Dense Homogeneous spaces Jan van Mill University of

Some results and problems on Countable Dense Homogeneous spaces Jan van Mill University of

Some results and problems on Countable Dense Homogeneous spaces Some results and problems on Countable Dense Homogeneous spaces Jan van Mill University of Amsterdam TU Delft Twelfth Symposium on General Topology and its Relations to Modern

1.08k views • 93 slides
Introduction to Parallel Computing George Karypis Dense Matrix Algorithms Outline Focus on

Introduction to Parallel Computing George Karypis Dense Matrix Algorithms Outline Focus on

Introduction to Parallel Computing George Karypis Dense Matrix Algorithms Outline Focus on numerical algorithms involving dense matrices: Matrix-Vector Multiplication Matrix-Matrix Multiplication Gaussian Elimination

734 views • 23 slides
Graph Resistance and Learning from Pairwise Comparisons pairwise comparisons of items. In

Graph Resistance and Learning from Pairwise Comparisons pairwise comparisons of items. In

Alex Olshevsky Department of ECE, Boston University Joint work with Julien Hendrickx (UC Louvain) and Venkatesh Saligrama (BU) Graph Resistance and Learning from Pairwise Comparisons pairwise comparisons of items. In many contexts,

759 views • 50 slides
K meson in dense matter L. Tols 1 , R. Molina 2 , E. Oset 2 and A. Ramos 3 1 Theory

K meson in dense matter L. Tols 1 , R. Molina 2 , E. Oset 2 and A. Ramos 3 1 Theory

K self-energy from the s-wave K K N interaction K in dense matter Nuclear transparency in the A Introduction Results K meson in dense matter L. Tols 1 , R. Molina 2 , E. Oset 2 and A. Ramos 3 1 Theory

494 views • 20 slides
Why Are We Matrices? Studying plenty Matrices have uses in Computer Science. E.g.: of

Why Are We Matrices? Studying plenty Matrices have uses in Computer Science. E.g.: of

Topic 7: Matrices - p. - CSc (Mccann) Matrlcs 245 v1.1 1 Why Are We Matrices? Studying plenty Matrices have uses in Computer Science. E.g.: of o Representation... o . graph (see . . of the data structure CSc

253 views • 8 slides
Transformations and Matrices Transformations I Transformations are functions Matrices

Transformations and Matrices Transformations I Transformations are functions Matrices

Transformations and Matrices Transformations I Transformations are functions Matrices are functions representations Matrices represent linear transfs CS5600 Computer Graphics Lecture Set 5 by 2 x 2 Matrices == 2D Linear

269 views • 9 slides
Selected results from trace gas inter-comparisons between AGAGE in situ and NOAA flask data P.B.

Selected results from trace gas inter-comparisons between AGAGE in situ and NOAA flask data P.B.

Selected results from trace gas inter-comparisons between AGAGE in situ and NOAA flask data P.B. Krummel 1 , L.P. Steele 1 , P.J. Fraser 1 , L.W. Porter 1* , N. Derek 1 , S.A. Montzka 2 , E.J. Dlugokencky 2 , G.S. Dutton 2 , B.D. Hall 2 , J.W.

411 views • 13 slides
Correction for multiple comparisons in FreeSurfer 1 Problem of Multiple Comparisons p < 10 -7

Correction for multiple comparisons in FreeSurfer 1 Problem of Multiple Comparisons p < 10 -7

Correction for multiple comparisons in FreeSurfer 1 Problem of Multiple Comparisons p < 10 -7 p < 0.01 p < 0.10 p value is probability that a voxel is falsely activated Threshold too liberal: many false positives Threshold too

230 views • 21 slides
Semimonotone Matrices Megan Wendler May 27, 2018 Megan Wendler Semimonotone Matrices May 27,

Semimonotone Matrices Megan Wendler May 27, 2018 Megan Wendler Semimonotone Matrices May 27,

Semimonotone Matrices Megan Wendler May 27, 2018 Megan Wendler Semimonotone Matrices May 27, 2018 1 / 37 Table of contents Introduction 1 The definition of semimonotone & an example Some observations and previous results Questions

1.1k views • 88 slides
BMI-206 Structure-Structure comparisons Sequence-Structure comparisons Marc A. Marti-Renom

BMI-206 Structure-Structure comparisons Sequence-Structure comparisons Marc A. Marti-Renom

BMI-206 Structure-Structure comparisons Sequence-Structure comparisons Marc A. Marti-Renom Assistant Adjunct Professor Department of Biopharmaceutical Sciences February 3rd, 2005 BMI206 01/19/2005 How to use this lectures Ask! Outline

1.1k views • 95 slides
Dense Flow Visualization Lecture 10 February 27, 2020 General Overview Dense methods in 2D

Dense Flow Visualization Lecture 10 February 27, 2020 General Overview Dense methods in 2D

CS53000 - Spring 2020 Introduction to Scientific Visualization Dense Flow Visualization Lecture 10 February 27, 2020 General Overview Dense methods in 2D Dense methods in 3D Dense methods on curved surfaces CS530 / Spring 2020 :

1.17k views • 73 slides
Adjacency Matrices Representations memory? 1. Adjacency matrices. 2. Adjacency lists. 3.

Adjacency Matrices Representations memory? 1. Adjacency matrices. 2. Adjacency lists. 3.

Lecture 11 | Part 1 Adjacency Matrices Representations memory? 1. Adjacency matrices. 2. Adjacency lists. 3. Dictionary of sets How do we store a graph in a computers Three approaches: Adjacency Matrices Assume nodes are

927 views • 26 slides
Dense cold mixes: Preservation of Dense cold mixes: Preservation of county roads county roads

Dense cold mixes: Preservation of Dense cold mixes: Preservation of county roads county roads

Dense cold mixes: Preservation of Dense cold mixes: Preservation of county roads county roads Eric Jorda Centre de Recherche Rhne-Alpes E. JORDA / Arkema / AEMA-ISAET September 2008 Dense cold mixes Definition : Emulsion mix

442 views • 17 slides
JUST THE MATHS SLIDES NUMBER 9.9 MATRICES 9 (Modal & spectral matrices) by

JUST THE MATHS SLIDES NUMBER 9.9 MATRICES 9 (Modal & spectral matrices) by

JUST THE MATHS SLIDES NUMBER 9.9 MATRICES 9 (Modal & spectral matrices) by A.J.Hobson 9.9.1 Assumptions and definitions 9.9.2 Diagonalisation of a matrix UNIT 9.9 - MATRICES 9 MODAL AND SPECTRAL MATRICES 9.9.1 ASSUMPTIONS AND

549 views • 10 slides
Square Matrices The first part deals with some notions and results from ring geometry and the

Square Matrices The first part deals with some notions and results from ring geometry and the

Geometries on -Hermitian Matrices Modern Algebra and Its Applications Batumi Rustaveli State University, Batumi, Georgia September 22nd, 2010 Joint work with Andrea Blunck (Hamburg, Germany) H ANS H AVLICEK F ORSCHUNGSGRUPPE D

471 views • 26 slides

More recommend