Lecture 02 Linear Algebra Basics

Size: px
Start display at page:

Download "Lecture 02 Linear Algebra Basics"

Transcription

1 Introduction to Computational Data Analysis CX4240, 2019 Spring Lecture 02 Linear Algebra Basics Chao Zhang College of Computing Georgia Tech These slides are based on slides from Le Song and Andres Mendez-Vazquez.

2 Outline Linear Algebra Basics Norms Multiplications Matrix Inversion Trace and Determinant Eigen Values and Eigen Vectors Singular Value Decomposition Matrix Calculus 2

3 Why Linear Algebra?

4 Linear Algebra Basics 4

5 Outline Linear Algebra Basics Norms Multiplications Matrix Inversion Trace and Determinant Eigen Values and Eigen Vectors Singular Value Decomposition Matrix Calculus 5

6 Norms 6

7 Norms 7

8 Vector Norm Examples 8

9 Special Matrices 9

10 Outline Linear Algebra Basics Norms Multiplications Matrix Inversion Trace and Determinant Eigen Values and Eigen Vectors Singular Value Decomposition Matrix Calculus 10

11 Multiplications 11

12 Multiplications 12

13 Inner Product Properties 13

14 Inner Product Properties 14

15 Inner Product Properties 15

16 Outline Linear Algebra Basics Norms Multiplications Matrix Inversion Trace and Determinant Eigen Values and Eigen Vectors Matrix Decomposition Matrix Calculus 16

17 Linear Independence and Matrix Rank 17

18 Range and Null Space 18

19 Row and Column Space 19

20 Matrix Rank: Examples What are the ranks for the following matrices? 20

21 Matrix Inverse 21

22 Outline Linear Algebra Basics Norms Multiplications Matrix Inversion Trace and Determinant Eigen Values and Eigen Vectors Singular Value Decomposition Matrix Calculus 22

23 Matrix Trace 23

24 Matrix Determinant 24

25 Properties of Matrix Determinant 25

26 Outline Linear Algebra Basics Norms Multiplications Matrix Inversion Trace and Determinant Eigen Values and Eigen Vectors Singular Value Decomposition Matrix Calculus 26

27 Eigenvalues and Eigenvectors 27

28 Computing Eigenvalues and Eigenvectors 28

29 Eigenvalue Example Slide credit: Shubham Kumbhar 29

30 Matrix Eigen Decomposition 30

31 Properties of Eigendecomposition 31

32 Outline Linear Algebra Basics Norms Multiplications Matrix Inversion Trace and Determinant Eigen Values and Eigen Vectors Singular Value Decomposition Matrix Calculus 32

33 Singular Value Decomposition 33

34 Singular Value Decomposition 34

35 Geometric Meaning of SVD Image Credit: Kevin Binz 35

36 M<latexit sha1_base64="fsxrvhuf+uha5zzl6jy675pf76s=">aaab8xicbvbns8nafhypx7v+vt16wsycp5kiomeify9cbvulbsib7uu7dlmjuxuhhp4llx4u8eq/8ea/cdpmok0dc8pme+y8crlbtxhdb6e0srq2vlhergxt7+zuvfcp2jpofcmwi0wsoghvkljelufgycdrskna4emwvs79hydumsfy3kws9cm6ldzkjborpfyiakzbmn1o+9waw3dnimvek0gncjt71a/eigzphniwqbxuem5i/iwqw5naaawxakwog9mhdi2vneltz7peu3jilqejy2wfngsm/t7iakt1jarszj5ql3q5+j/xtu146wdcjqlbyeyfhakgjib5+wtaftijjpzqprjnstiiksqmlalis/awt14m7bo659a9u/na46qoowxhcayn4mefnoagmtacbhke4rxeho28oo/ox3y05bq7h/ahzucpu+kq8q==</latexit> <latexit sha1_base64="fsxrvhuf+uha5zzl6jy675pf76s=">aaab8xicbvbns8nafhypx7v+vt16wsycp5kiomeify9cbvulbsib7uu7dlmjuxuhhp4llx4u8eq/8ea/cdpmok0dc8pme+y8crlbtxhdb6e0srq2vlhergxt7+zuvfcp2jpofcmwi0wsoghvkljelufgycdrskna4emwvs79hydumsfy3kws9cm6ldzkjborpfyiakzbmn1o+9waw3dnimvek0gncjt71a/eigzphniwqbxuem5i/iwqw5naaawxakwog9mhdi2vneltz7peu3jilqejy2wfngsm/t7iakt1jarszj5ql3q5+j/xtu146wdcjqlbyeyfhakgjib5+wtaftijjpzqprjnstiiksqmlalis/awt14m7bo659a9u/na46qoowxhcayn4mefnoagmtacbhke4rxeho28oo/ox3y05bq7h/ahzucpu+kq8q==</latexit> <latexit sha1_base64="fsxrvhuf+uha5zzl6jy675pf76s=">aaab8xicbvbns8nafhypx7v+vt16wsycp5kiomeify9cbvulbsib7uu7dlmjuxuhhp4llx4u8eq/8ea/cdpmok0dc8pme+y8crlbtxhdb6e0srq2vlhergxt7+zuvfcp2jpofcmwi0wsoghvkljelufgycdrskna4emwvs79hydumsfy3kws9cm6ldzkjborpfyiakzbmn1o+9waw3dnimvek0gncjt71a/eigzphniwqbxuem5i/iwqw5naaawxakwog9mhdi2vneltz7peu3jilqejy2wfngsm/t7iakt1jarszj5ql3q5+j/xtu146wdcjqlbyeyfhakgjib5+wtaftijjpzqprjnstiiksqmlalis/awt14m7bo659a9u/na46qoowxhcayn4mefnoagmtacbhke4rxeho28oo/ox3y05bq7h/ahzucpu+kq8q==</latexit> <latexit sha1_base64="fsxrvhuf+uha5zzl6jy675pf76s=">aaab8xicbvbns8nafhypx7v+vt16wsycp5kiomeify9cbvulbsib7uu7dlmjuxuhhp4llx4u8eq/8ea/cdpmok0dc8pme+y8crlbtxhdb6e0srq2vlhergxt7+zuvfcp2jpofcmwi0wsoghvkljelufgycdrskna4emwvs79hydumsfy3kws9cm6ldzkjborpfyiakzbmn1o+9waw3dnimvek0gncjt71a/eigzphniwqbxuem5i/iwqw5naaawxakwog9mhdi2vneltz7peu3jilqejy2wfngsm/t7iakt1jarszj5ql3q5+j/xtu146wdcjqlbyeyfhakgjib5+wtaftijjpzqprjnstiiksqmlalis/awt14m7bo659a9u/na46qoowxhcayn4mefnoagmtacbhke4rxeho28oo/ox3y05bq7h/ahzucpu+kq8q==</latexit> U<latexit sha1_base64="i69u2bbvulryn7n5sm3b9sq3yd8=">aaab8xicbvbns8nafhypx7v+vt16wsycp5kiomeif48vtftss9lsn+3szsbsvggl9f948aciv/+nn/+nmzyhbr1yggbey+dnkehh0hw/ndla+sbmvnm7sro7t39qptxqmtjvjpsslrhubnrwkrt3uadknurzggwst4pjbe63n7g2ilypoe14p6ijjulbkfrpsrdrhadh5s8g1zpbd+cgq8qrsa0knafvr94wzmneftjjjel6bol9jgoutpjzpzcanla2ospetvtrijt+nk88i2dwgziw1vypjhp190zgi2omuwan84rm2cvf/7xuiuf1pxmqszertvgotcxbmotnk6hqnkgcwkkzfjyrywoqkunbusww4c2fvepaf3xprxv3l7xgtvfhgu7gfm7bgytowb00wqcgcp7hfd4c47w4787hyrtkfdvh8afo5w/icpd5</latexit> <latexit sha1_base64="i69u2bbvulryn7n5sm3b9sq3yd8=">aaab8xicbvbns8nafhypx7v+vt16wsycp5kiomeif48vtftss9lsn+3szsbsvggl9f948aciv/+nn/+nmzyhbr1yggbey+dnkehh0hw/ndla+sbmvnm7sro7t39qptxqmtjvjpsslrhubnrwkrt3uadknurzggwst4pjbe63n7g2ilypoe14p6ijjulbkfrpsrdrhadh5s8g1zpbd+cgq8qrsa0knafvr94wzmneftjjjel6bol9jgoutpjzpzcanla2ospetvtrijt+nk88i2dwgziw1vypjhp190zgi2omuwan84rm2cvf/7xuiuf1pxmqszertvgotcxbmotnk6hqnkgcwkkzfjyrywoqkunbusww4c2fvepaf3xprxv3l7xgtvfhgu7gfm7bgytowb00wqcgcp7hfd4c47w4787hyrtkfdvh8afo5w/icpd5</latexit> <latexit sha1_base64="i69u2bbvulryn7n5sm3b9sq3yd8=">aaab8xicbvbns8nafhypx7v+vt16wsycp5kiomeif48vtftss9lsn+3szsbsvggl9f948aciv/+nn/+nmzyhbr1yggbey+dnkehh0hw/ndla+sbmvnm7sro7t39qptxqmtjvjpsslrhubnrwkrt3uadknurzggwst4pjbe63n7g2ilypoe14p6ijjulbkfrpsrdrhadh5s8g1zpbd+cgq8qrsa0knafvr94wzmneftjjjel6bol9jgoutpjzpzcanla2ospetvtrijt+nk88i2dwgziw1vypjhp190zgi2omuwan84rm2cvf/7xuiuf1pxmqszertvgotcxbmotnk6hqnkgcwkkzfjyrywoqkunbusww4c2fvepaf3xprxv3l7xgtvfhgu7gfm7bgytowb00wqcgcp7hfd4c47w4787hyrtkfdvh8afo5w/icpd5</latexit> <latexit sha1_base64="i69u2bbvulryn7n5sm3b9sq3yd8=">aaab8xicbvbns8nafhypx7v+vt16wsycp5kiomeif48vtftss9lsn+3szsbsvggl9f948aciv/+nn/+nmzyhbr1yggbey+dnkehh0hw/ndla+sbmvnm7sro7t39qptxqmtjvjpsslrhubnrwkrt3uadknurzggwst4pjbe63n7g2ilypoe14p6ijjulbkfrpsrdrhadh5s8g1zpbd+cgq8qrsa0knafvr94wzmneftjjjel6bol9jgoutpjzpzcanla2ospetvtrijt+nk88i2dwgziw1vypjhp190zgi2omuwan84rm2cvf/7xuiuf1pxmqszertvgotcxbmotnk6hqnkgcwkkzfjyrywoqkunbusww4c2fvepaf3xprxv3l7xgtvfhgu7gfm7bgytowb00wqcgcp7hfd4c47w4787hyrtkfdvh8afo5w/icpd5</latexit> <latexit sha1_base64="uqhygdb8ht/vcx5pjpqswfwf/lk=">aaab+hicbvdlsgmxfl1tx7u+wnxpjlgev2vgbf0w3bisab/qguomzbshswzimkid+ivuxcji1k9x59+yawehrqcch3pu5z6cmofmg9f9dkpr6xubw+xtys7u3n61dndy0xgqcg2tmmeqf2jnozo0bzjhtjcoikxiatec3or+95eqzwl5ykyjdqqesryxgo2vbrwql7azh1hm37orwlnbre423dnqkvekuoccruhtyx/gjbvugskx1n3ptuyqywuy4xrw8vnne0wmeet7lkosqa6yefazorxkeewxsk8anfd/b2ryad0voz3my+pllxf/8/qpia6cjmkknvssxaeo5cjekg8bdzmixpcpjzgozrmimsyke2o7qtgsvouvr5loecnzg97drb15xdrrhmm4gtpw4bkacastaaobfj7hfd6cj+ffexc+fqmlp9g5gj9wpn8adl+tva==</latexit> <latexit sha1_base64="uqhygdb8ht/vcx5pjpqswfwf/lk=">aaab+hicbvdlsgmxfl1tx7u+wnxpjlgev2vgbf0w3bisab/qguomzbshswzimkid+ivuxcji1k9x59+yawehrqcch3pu5z6cmofmg9f9dkpr6xubw+xtys7u3n61dndy0xgqcg2tmmeqf2jnozo0bzjhtjcoikxiatec3or+95eqzwl5ykyjdqqesryxgo2vbrwql7azh1hm37orwlnbre423dnqkvekuoccruhtyx/gjbvugskx1n3ptuyqywuy4xrw8vnne0wmeet7lkosqa6yefazorxkeewxsk8anfd/b2ryad0voz3my+pllxf/8/qpia6cjmkknvssxaeo5cjekg8bdzmixpcpjzgozrmimsyke2o7qtgsvouvr5loecnzg97drb15xdrrhmm4gtpw4bkacastaaobfj7hfd6cj+ffexc+fqmlp9g5gj9wpn8adl+tva==</latexit> <latexit sha1_base64="uqhygdb8ht/vcx5pjpqswfwf/lk=">aaab+hicbvdlsgmxfl1tx7u+wnxpjlgev2vgbf0w3bisab/qguomzbshswzimkid+ivuxcji1k9x59+yawehrqcch3pu5z6cmofmg9f9dkpr6xubw+xtys7u3n61dndy0xgqcg2tmmeqf2jnozo0bzjhtjcoikxiatec3or+95eqzwl5ykyjdqqesryxgo2vbrwql7azh1hm37orwlnbre423dnqkvekuoccruhtyx/gjbvugskx1n3ptuyqywuy4xrw8vnne0wmeet7lkosqa6yefazorxkeewxsk8anfd/b2ryad0voz3my+pllxf/8/qpia6cjmkknvssxaeo5cjekg8bdzmixpcpjzgozrmimsyke2o7qtgsvouvr5loecnzg97drb15xdrrhmm4gtpw4bkacastaaobfj7hfd6cj+ffexc+fqmlp9g5gj9wpn8adl+tva==</latexit> <latexit sha1_base64="uqhygdb8ht/vcx5pjpqswfwf/lk=">aaab+hicbvdlsgmxfl1tx7u+wnxpjlgev2vgbf0w3bisab/qguomzbshswzimkid+ivuxcji1k9x59+yawehrqcch3pu5z6cmofmg9f9dkpr6xubw+xtys7u3n61dndy0xgqcg2tmmeqf2jnozo0bzjhtjcoikxiatec3or+95eqzwl5ykyjdqqesryxgo2vbrwql7azh1hm37orwlnbre423dnqkvekuoccruhtyx/gjbvugskx1n3ptuyqywuy4xrw8vnne0wmeet7lkosqa6yefazorxkeewxsk8anfd/b2ryad0voz3my+pllxf/8/qpia6cjmkknvssxaeo5cjekg8bdzmixpcpjzgozrmimsyke2o7qtgsvouvr5loecnzg97drb15xdrrhmm4gtpw4bkacastaaobfj7hfd6cj+ffexc+fqmlp9g5gj9wpn8adl+tva==</latexit> <latexit sha1_base64="2ehdqpzqsme6l8h0i0lcyad4h94=">aaab/hicbvdlssnafj3uv62vajdugkvwvrirdfl047jcx9demplo2qgtszi5euoiv+lghsju/rb3/o2tngttptbwoode7pnjx5wpso1vo7k2vrg5vd2u7ezu7r+yh0c9fsws0c6jecqhplaum0g7widtqswpdn1o+/7spvd7d1qqfokopdh1qjwrlgaeg5zgzt0nmuz9iovl9y7qr8g6+chs2e17dmuvocvpobltkfnljioshfqa4vipowph4gvyaioc5ju3uttgziyndkipwcfvxjypn1unwhlbqst1e2dn1d8bgq6vsknftxzr1bjxip95wwscky9jik6acri4fctcgsgqmrdgtficpnuee8l0votmscqedf81xykz/ovv0jtvonbtubtotk7loqrogj2gm+sgs9rct6inuoigfd2jv/rmpbkvxrvxsritgovohf2b8fkdgnmvua==</latexit> <latexit sha1_base64="2ehdqpzqsme6l8h0i0lcyad4h94=">aaab/hicbvdlssnafj3uv62vajdugkvwvrirdfl047jcx9demplo2qgtszi5euoiv+lghsju/rb3/o2tngttptbwoode7pnjx5wpso1vo7k2vrg5vd2u7ezu7r+yh0c9fsws0c6jecqhplaum0g7widtqswpdn1o+/7spvd7d1qqfokopdh1qjwrlgaeg5zgzt0nmuz9iovl9y7qr8g6+chs2e17dmuvocvpobltkfnljioshfqa4vipowph4gvyaioc5ju3uttgziyndkipwcfvxjypn1unwhlbqst1e2dn1d8bgq6vsknftxzr1bjxip95wwscky9jik6acri4fctcgsgqmrdgtficpnuee8l0votmscqedf81xykz/ovv0jtvonbtubtotk7loqrogj2gm+sgs9rct6inuoigfd2jv/rmpbkvxrvxsritgovohf2b8fkdgnmvua==</latexit> <latexit sha1_base64="2ehdqpzqsme6l8h0i0lcyad4h94=">aaab/hicbvdlssnafj3uv62vajdugkvwvrirdfl047jcx9demplo2qgtszi5euoiv+lghsju/rb3/o2tngttptbwoode7pnjx5wpso1vo7k2vrg5vd2u7ezu7r+yh0c9fsws0c6jecqhplaum0g7widtqswpdn1o+/7spvd7d1qqfokopdh1qjwrlgaeg5zgzt0nmuz9iovl9y7qr8g6+chs2e17dmuvocvpobltkfnljioshfqa4vipowph4gvyaioc5ju3uttgziyndkipwcfvxjypn1unwhlbqst1e2dn1d8bgq6vsknftxzr1bjxip95wwscky9jik6acri4fctcgsgqmrdgtficpnuee8l0votmscqedf81xykz/ovv0jtvonbtubtotk7loqrogj2gm+sgs9rct6inuoigfd2jv/rmpbkvxrvxsritgovohf2b8fkdgnmvua==</latexit> <latexit sha1_base64="2ehdqpzqsme6l8h0i0lcyad4h94=">aaab/hicbvdlssnafj3uv62vajdugkvwvrirdfl047jcx9demplo2qgtszi5euoiv+lghsju/rb3/o2tngttptbwoode7pnjx5wpso1vo7k2vrg5vd2u7ezu7r+yh0c9fsws0c6jecqhplaum0g7widtqswpdn1o+/7spvd7d1qqfokopdh1qjwrlgaeg5zgzt0nmuz9iovl9y7qr8g6+chs2e17dmuvocvpobltkfnljioshfqa4vipowph4gvyaioc5ju3uttgziyndkipwcfvxjypn1unwhlbqst1e2dn1d8bgq6vsknftxzr1bjxip95wwscky9jik6acri4fctcgsgqmrdgtficpnuee8l0votmscqedf81xykz/ovv0jtvonbtubtotk7loqrogj2gm+sgs9rct6inuoigfd2jv/rmpbkvxrvxsritgovohf2b8fkdgnmvua==</latexit> SVD Example V T From [Strang] 36

37 Outline Linear Algebra Basics Norms Multiplications Matrix Inversion Trace and Determinant Eigen Values and Eigen Vectors Singular Value Decomposition Matrix Calculus 37

38 <latexit sha1_base64="11y8g8fa9znqeubrie41aw9zolk=">aaab6nicbvbns8naej3ur1q/qh69lbbbu0leqmeif48v7qe0owy2m3bpzhn2j2ij/qlepcji1v/kzx/jts1bwx8mpn6bywzekehh0hw/ncla+sbmvng7tlo7t39qpjxqmtjvjddzlgpdcajhuijerigsdxlnarri3g7gnzo//ci1ebf6wenc/ygolqgfo2il+6f+uf+uufv3drjkvjxuieejx/7qdwkwrlwhk9syrucm6gduo2cst0u91pcesjed8q6likbc+nn81ck5s8qahlg2pzdm1d8tgy2mmusb7ywojsyynxp/87ophld+jlssildssshmjcgyzp4ma6e5qzmxhdit7k2ejaimdg06jruct/zykmldvd236t1dvurxerxfoiftoacpalchw2haexgm4rle4c2rzovz7nwswgtopnmmf+b8/gbjmi3a</latexit> <latexit sha1_base64="11y8g8fa9znqeubrie41aw9zolk=">aaab6nicbvbns8naej3ur1q/qh69lbbbu0leqmeif48v7qe0owy2m3bpzhn2j2ij/qlepcji1v/kzx/jts1bwx8mpn6bywzekehh0hw/ncla+sbmvng7tlo7t39qpjxqmtjvjddzlgpdcajhuijerigsdxlnarri3g7gnzo//ci1ebf6wenc/ygolqgfo2il+6f+uf+uufv3drjkvjxuieejx/7qdwkwrlwhk9syrucm6gduo2cst0u91pcesjed8q6likbc+nn81ck5s8qahlg2pzdm1d8tgy2mmusb7ywojsyynxp/87ophld+jlssildssshmjcgyzp4ma6e5qzmxhdit7k2ejaimdg06jruct/zykmldvd236t1dvurxerxfoiftoacpalchw2haexgm4rle4c2rzovz7nwswgtopnmmf+b8/gbjmi3a</latexit> <latexit sha1_base64="11y8g8fa9znqeubrie41aw9zolk=">aaab6nicbvbns8naej3ur1q/qh69lbbbu0leqmeif48v7qe0owy2m3bpzhn2j2ij/qlepcji1v/kzx/jts1bwx8mpn6bywzekehh0hw/ncla+sbmvng7tlo7t39qpjxqmtjvjddzlgpdcajhuijerigsdxlnarri3g7gnzo//ci1ebf6wenc/ygolqgfo2il+6f+uf+uufv3drjkvjxuieejx/7qdwkwrlwhk9syrucm6gduo2cst0u91pcesjed8q6likbc+nn81ck5s8qahlg2pzdm1d8tgy2mmusb7ywojsyynxp/87ophld+jlssildssshmjcgyzp4ma6e5qzmxhdit7k2ejaimdg06jruct/zykmldvd236t1dvurxerxfoiftoacpalchw2haexgm4rle4c2rzovz7nwswgtopnmmf+b8/gbjmi3a</latexit> <latexit sha1_base64="11y8g8fa9znqeubrie41aw9zolk=">aaab6nicbvbns8naej3ur1q/qh69lbbbu0leqmeif48v7qe0owy2m3bpzhn2j2ij/qlepcji1v/kzx/jts1bwx8mpn6bywzekehh0hw/ncla+sbmvng7tlo7t39qpjxqmtjvjddzlgpdcajhuijerigsdxlnarri3g7gnzo//ci1ebf6wenc/ygolqgfo2il+6f+uf+uufv3drjkvjxuieejx/7qdwkwrlwhk9syrucm6gduo2cst0u91pcesjed8q6likbc+nn81ck5s8qahlg2pzdm1d8tgy2mmusb7ywojsyynxp/87ophld+jlssildssshmjcgyzp4ma6e5qzmxhdit7k2ejaimdg06jruct/zykmldvd236t1dvurxerxfoiftoacpalchw2haexgm4rle4c2rzovz7nwswgtopnmmf+b8/gbjmi3a</latexit> Matrix Calculus x k 38

39 Summary Linear Algebra Basics Norms Multiplications Matrix Inversion Trace and Determinant Eigen Values and Eigen Vectors Singular Value Decomposition Matrix Calculus 39

Singular Value Decomposition

Singular Value Decomposition Singular Value Decomposition Motivatation The diagonalization theorem play a part in many interesting applications. Unfortunately not all matrices can be factored as A = PDP However a factorization A =

More information

Multivariate Statistical Analysis

Multivariate Statistical Analysis Multivariate Statistical Analysis Fall 2011 C. L. Williams, Ph.D. Lecture 4 for Applied Multivariate Analysis Outline 1 Eigen values and eigen vectors Characteristic equation Some properties of eigendecompositions

More information

Background Mathematics (2/2) 1. David Barber

Background Mathematics (2/2) 1. David Barber Background Mathematics (2/2) 1 David Barber University College London Modified by Samson Cheung (sccheung@ieee.org) 1 These slides accompany the book Bayesian Reasoning and Machine Learning. The book and

More information

Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2

Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2 Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition

More information

Image Registration Lecture 2: Vectors and Matrices

Image Registration Lecture 2: Vectors and Matrices Image Registration Lecture 2: Vectors and Matrices Prof. Charlene Tsai Lecture Overview Vectors Matrices Basics Orthogonal matrices Singular Value Decomposition (SVD) 2 1 Preliminary Comments Some of this

More information

EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016

EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016 EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016 Answer the questions in the spaces provided on the question sheets. You must show your work to get credit for your answers. There will

More information

Linear Algebra Review. Vectors

Linear Algebra Review. Vectors Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors

More information

Example Linear Algebra Competency Test

Example Linear Algebra Competency Test Example Linear Algebra Competency Test The 4 questions below are a combination of True or False, multiple choice, fill in the blank, and computations involving matrices and vectors. In the latter case,

More information

Foundations of Computer Vision

Foundations of Computer Vision Foundations of Computer Vision Wesley. E. Snyder North Carolina State University Hairong Qi University of Tennessee, Knoxville Last Edited February 8, 2017 1 3.2. A BRIEF REVIEW OF LINEAR ALGEBRA Apply

More information

Review of Some Concepts from Linear Algebra: Part 2

Review of Some Concepts from Linear Algebra: Part 2 Review of Some Concepts from Linear Algebra: Part 2 Department of Mathematics Boise State University January 16, 2019 Math 566 Linear Algebra Review: Part 2 January 16, 2019 1 / 22 Vector spaces A set

More information

Large Scale Data Analysis Using Deep Learning

Large Scale Data Analysis Using Deep Learning Large Scale Data Analysis Using Deep Learning Linear Algebra U Kang Seoul National University U Kang 1 In This Lecture Overview of linear algebra (but, not a comprehensive survey) Focused on the subset

More information

HW2 - Due 01/30. Each answer must be mathematically justified. Don t forget your name.

HW2 - Due 01/30. Each answer must be mathematically justified. Don t forget your name. HW2 - Due 0/30 Each answer must be mathematically justified. Don t forget your name. Problem. Use the row reduction algorithm to find the inverse of the matrix 0 0, 2 3 5 if it exists. Double check your

More information

Maths for Signals and Systems Linear Algebra in Engineering

Maths for Signals and Systems Linear Algebra in Engineering Maths for Signals and Systems Linear Algebra in Engineering Lectures 13 15, Tuesday 8 th and Friday 11 th November 016 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE

More information

Lecture II: Linear Algebra Revisited

Lecture II: Linear Algebra Revisited Lecture II: Linear Algebra Revisited Overview Vector spaces, Hilbert & Banach Spaces, etrics & Norms atrices, Eigenvalues, Orthogonal Transformations, Singular Values Operators, Operator Norms, Function

More information

Applied Linear Algebra in Geoscience Using MATLAB

Applied Linear Algebra in Geoscience Using MATLAB Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in

More information

1 Linearity and Linear Systems

1 Linearity and Linear Systems Mathematical Tools for Neuroscience (NEU 34) Princeton University, Spring 26 Jonathan Pillow Lecture 7-8 notes: Linear systems & SVD Linearity and Linear Systems Linear system is a kind of mapping f( x)

More information

HOSTOS COMMUNITY COLLEGE DEPARTMENT OF MATHEMATICS

HOSTOS COMMUNITY COLLEGE DEPARTMENT OF MATHEMATICS HOSTOS COMMUNITY COLLEGE DEPARTMENT OF MATHEMATICS MAT 217 Linear Algebra CREDIT HOURS: 4.0 EQUATED HOURS: 4.0 CLASS HOURS: 4.0 PREREQUISITE: PRE/COREQUISITE: MAT 210 Calculus I MAT 220 Calculus II RECOMMENDED

More information

Linear Algebra & Geometry why is linear algebra useful in computer vision?

Linear Algebra & Geometry why is linear algebra useful in computer vision? Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia

More information

Data Mining Lecture 4: Covariance, EVD, PCA & SVD

Data Mining Lecture 4: Covariance, EVD, PCA & SVD Data Mining Lecture 4: Covariance, EVD, PCA & SVD Jo Houghton ECS Southampton February 25, 2019 1 / 28 Variance and Covariance - Expectation A random variable takes on different values due to chance The

More information

There are six more problems on the next two pages

There are six more problems on the next two pages Math 435 bg & bu: Topics in linear algebra Summer 25 Final exam Wed., 8/3/5. Justify all your work to receive full credit. Name:. Let A 3 2 5 Find a permutation matrix P, a lower triangular matrix L with

More information

CS 143 Linear Algebra Review

CS 143 Linear Algebra Review CS 143 Linear Algebra Review Stefan Roth September 29, 2003 Introductory Remarks This review does not aim at mathematical rigor very much, but instead at ease of understanding and conciseness. Please see

More information

Chap 3. Linear Algebra

Chap 3. Linear Algebra Chap 3. Linear Algebra Outlines 1. Introduction 2. Basis, Representation, and Orthonormalization 3. Linear Algebraic Equations 4. Similarity Transformation 5. Diagonal Form and Jordan Form 6. Functions

More information

Linear Algebra, part 3 QR and SVD

Linear Algebra, part 3 QR and SVD Linear Algebra, part 3 QR and SVD Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2012 Going back to least squares (Section 1.4 from Strang, now also see section 5.2). We

More information

Fall TMA4145 Linear Methods. Exercise set Given the matrix 1 2

Fall TMA4145 Linear Methods. Exercise set Given the matrix 1 2 Norwegian University of Science and Technology Department of Mathematical Sciences TMA445 Linear Methods Fall 07 Exercise set Please justify your answers! The most important part is how you arrive at an

More information

Review of Linear Algebra

Review of Linear Algebra Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=

More information

COMP 558 lecture 18 Nov. 15, 2010

COMP 558 lecture 18 Nov. 15, 2010 Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to

More information

Review problems for MA 54, Fall 2004.

Review problems for MA 54, Fall 2004. Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on

More information

EIGENVALUES AND SINGULAR VALUE DECOMPOSITION

EIGENVALUES AND SINGULAR VALUE DECOMPOSITION APPENDIX B EIGENVALUES AND SINGULAR VALUE DECOMPOSITION B.1 LINEAR EQUATIONS AND INVERSES Problems of linear estimation can be written in terms of a linear matrix equation whose solution provides the required

More information

Review of Linear Algebra

Review of Linear Algebra Review of Linear Algebra Dr Gerhard Roth COMP 40A Winter 05 Version Linear algebra Is an important area of mathematics It is the basis of computer vision Is very widely taught, and there are many resources

More information

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v

More information

Special Two-Semester Linear Algebra Course (Fall 2012 and Spring 2013)

Special Two-Semester Linear Algebra Course (Fall 2012 and Spring 2013) Special Two-Semester Linear Algebra Course (Fall 2012 and Spring 2013) The first semester will concentrate on basic matrix skills as described in MA 205, and the student should have one semester of calculus.

More information

Linear Algebra - Part II

Linear Algebra - Part II Linear Algebra - Part II Projection, Eigendecomposition, SVD (Adapted from Sargur Srihari s slides) Brief Review from Part 1 Symmetric Matrix: A = A T Orthogonal Matrix: A T A = AA T = I and A 1 = A T

More information

Latent Semantic Analysis. Hongning Wang

Latent Semantic Analysis. Hongning Wang Latent Semantic Analysis Hongning Wang CS@UVa VS model in practice Document and query are represented by term vectors Terms are not necessarily orthogonal to each other Synonymy: car v.s. automobile Polysemy:

More information

Linear Algebra & Geometry why is linear algebra useful in computer vision?

Linear Algebra & Geometry why is linear algebra useful in computer vision? Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia

More information

2. Review of Linear Algebra

2. Review of Linear Algebra 2. Review of Linear Algebra ECE 83, Spring 217 In this course we will represent signals as vectors and operators (e.g., filters, transforms, etc) as matrices. This lecture reviews basic concepts from linear

More information

14 Singular Value Decomposition

14 Singular Value Decomposition 14 Singular Value Decomposition For any high-dimensional data analysis, one s first thought should often be: can I use an SVD? The singular value decomposition is an invaluable analysis tool for dealing

More information

Lecture 9: SVD, Low Rank Approximation

Lecture 9: SVD, Low Rank Approximation CSE 521: Design and Analysis of Algorithms I Spring 2016 Lecture 9: SVD, Low Rank Approimation Lecturer: Shayan Oveis Gharan April 25th Scribe: Koosha Khalvati Disclaimer: hese notes have not been subjected

More information

Lecture 8. Principal Component Analysis. Luigi Freda. ALCOR Lab DIAG University of Rome La Sapienza. December 13, 2016

Lecture 8. Principal Component Analysis. Luigi Freda. ALCOR Lab DIAG University of Rome La Sapienza. December 13, 2016 Lecture 8 Principal Component Analysis Luigi Freda ALCOR Lab DIAG University of Rome La Sapienza December 13, 2016 Luigi Freda ( La Sapienza University) Lecture 8 December 13, 2016 1 / 31 Outline 1 Eigen

More information

Lecture Note 1: Background

Lecture Note 1: Background ECE5463: Introduction to Robotics Lecture Note 1: Background Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture 1 (ECE5463 Sp18)

More information

Linear Algebra in Computer Vision. Lecture2: Basic Linear Algebra & Probability. Vector. Vector Operations

Linear Algebra in Computer Vision. Lecture2: Basic Linear Algebra & Probability. Vector. Vector Operations Linear Algebra in Computer Vision CSED441:Introduction to Computer Vision (2017F Lecture2: Basic Linear Algebra & Probability Bohyung Han CSE, POSTECH bhhan@postech.ac.kr Mathematics in vector space Linear

More information

Basic Calculus Review

Basic Calculus Review Basic Calculus Review Lorenzo Rosasco ISML Mod. 2 - Machine Learning Vector Spaces Functionals and Operators (Matrices) Vector Space A vector space is a set V with binary operations +: V V V and : R V

More information

Fundamentals of Matrices

Fundamentals of Matrices Maschinelles Lernen II Fundamentals of Matrices Christoph Sawade/Niels Landwehr/Blaine Nelson Tobias Scheffer Matrix Examples Recap: Data Linear Model: f i x = w i T x Let X = x x n be the data matrix

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) Lecture 1: Course Overview; Matrix Multiplication Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical

More information

10-701/ Recitation : Linear Algebra Review (based on notes written by Jing Xiang)

10-701/ Recitation : Linear Algebra Review (based on notes written by Jing Xiang) 10-701/15-781 Recitation : Linear Algebra Review (based on notes written by Jing Xiang) Manojit Nandi February 1, 2014 Outline Linear Algebra General Properties Matrix Operations Inner Products and Orthogonal

More information

Mathematical foundations - linear algebra

Mathematical foundations - linear algebra Mathematical foundations - linear algebra Andrea Passerini passerini@disi.unitn.it Machine Learning Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar

More information

Parallel Singular Value Decomposition. Jiaxing Tan

Parallel Singular Value Decomposition. Jiaxing Tan Parallel Singular Value Decomposition Jiaxing Tan Outline What is SVD? How to calculate SVD? How to parallelize SVD? Future Work What is SVD? Matrix Decomposition Eigen Decomposition A (non-zero) vector

More information

LAKELAND COMMUNITY COLLEGE COURSE OUTLINE FORM

LAKELAND COMMUNITY COLLEGE COURSE OUTLINE FORM LAKELAND COMMUNITY COLLEGE COURSE OUTLINE FORM ORIGINATION DATE: 8/2/99 APPROVAL DATE: 3/22/12 LAST MODIFICATION DATE: 3/28/12 EFFECTIVE TERM/YEAR: FALL/ 12 COURSE ID: COURSE TITLE: MATH2800 Linear Algebra

More information

CENTRAL TEXAS COLLEGE SYLLABUS FOR MATH 2318 Linear Algebra. Semester Hours Credit: 3

CENTRAL TEXAS COLLEGE SYLLABUS FOR MATH 2318 Linear Algebra. Semester Hours Credit: 3 CENTRAL TEXAS COLLEGE SYLLABUS FOR MATH 2318 Linear Algebra Semester Hours Credit: 3 I. INTRODUCTION A. Linear Algebra is a three semester-hour course. This course introduces and provides models for application

More information

15 Singular Value Decomposition

15 Singular Value Decomposition 15 Singular Value Decomposition For any high-dimensional data analysis, one s first thought should often be: can I use an SVD? The singular value decomposition is an invaluable analysis tool for dealing

More information

Singular Value Decomposition

Singular Value Decomposition Singular Value Decomposition (Com S 477/577 Notes Yan-Bin Jia Sep, 7 Introduction Now comes a highlight of linear algebra. Any real m n matrix can be factored as A = UΣV T where U is an m m orthogonal

More information

LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS

LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,

More information

CS 231A Section 1: Linear Algebra & Probability Review. Kevin Tang

CS 231A Section 1: Linear Algebra & Probability Review. Kevin Tang CS 231A Section 1: Linear Algebra & Probability Review Kevin Tang Kevin Tang Section 1-1 9/30/2011 Topics Support Vector Machines Boosting Viola Jones face detector Linear Algebra Review Notation Operations

More information

Review of similarity transformation and Singular Value Decomposition

Review of similarity transformation and Singular Value Decomposition Review of similarity transformation and Singular Value Decomposition Nasser M Abbasi Applied Mathematics Department, California State University, Fullerton July 8 7 page compiled on June 9, 5 at 9:5pm

More information

Conceptual Questions for Review

Conceptual Questions for Review Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.

More information

CAAM 335 Matrix Analysis

CAAM 335 Matrix Analysis CAAM 335 Matrix Analysis Solutions to Homework 8 Problem (5+5+5=5 points The partial fraction expansion of the resolvent for the matrix B = is given by (si B = s } {{ } =P + s + } {{ } =P + (s (5 points

More information

Linear Algebra, part 3. Going back to least squares. Mathematical Models, Analysis and Simulation = 0. a T 1 e. a T n e. Anna-Karin Tornberg

Linear Algebra, part 3. Going back to least squares. Mathematical Models, Analysis and Simulation = 0. a T 1 e. a T n e. Anna-Karin Tornberg Linear Algebra, part 3 Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2010 Going back to least squares (Sections 1.7 and 2.3 from Strang). We know from before: The vector

More information

10-725/36-725: Convex Optimization Prerequisite Topics

10-725/36-725: Convex Optimization Prerequisite Topics 10-725/36-725: Convex Optimization Prerequisite Topics February 3, 2015 This is meant to be a brief, informal refresher of some topics that will form building blocks in this course. The content of the

More information

Lecture 6 Positive Definite Matrices

Lecture 6 Positive Definite Matrices Linear Algebra Lecture 6 Positive Definite Matrices Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Spring 2017 2017/6/8 Lecture 6: Positive Definite Matrices

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) AMS526: Numerical Analysis (Numerical Linear Algebra for Computational and Data Sciences) Lecture 14: Eigenvalue Problems; Eigenvalue Revealing Factorizations Xiangmin Jiao Stony Brook University Xiangmin

More information

Final Exam, Linear Algebra, Fall, 2003, W. Stephen Wilson

Final Exam, Linear Algebra, Fall, 2003, W. Stephen Wilson Final Exam, Linear Algebra, Fall, 2003, W. Stephen Wilson Name: TA Name and section: NO CALCULATORS, SHOW ALL WORK, NO OTHER PAPERS ON DESK. There is very little actual work to be done on this exam if

More information

1. The Polar Decomposition

1. The Polar Decomposition A PERSONAL INTERVIEW WITH THE SINGULAR VALUE DECOMPOSITION MATAN GAVISH Part. Theory. The Polar Decomposition In what follows, F denotes either R or C. The vector space F n is an inner product space with

More information

Dimensionality Reduction: PCA. Nicholas Ruozzi University of Texas at Dallas

Dimensionality Reduction: PCA. Nicholas Ruozzi University of Texas at Dallas Dimensionality Reduction: PCA Nicholas Ruozzi University of Texas at Dallas Eigenvalues λ is an eigenvalue of a matrix A R n n if the linear system Ax = λx has at least one non-zero solution If Ax = λx

More information

Eigenvalues and diagonalization

Eigenvalues and diagonalization Eigenvalues and diagonalization Patrick Breheny November 15 Patrick Breheny BST 764: Applied Statistical Modeling 1/20 Introduction The next topic in our course, principal components analysis, revolves

More information

Lecture 13. Principal Component Analysis. Brett Bernstein. April 25, CDS at NYU. Brett Bernstein (CDS at NYU) Lecture 13 April 25, / 26

Lecture 13. Principal Component Analysis. Brett Bernstein. April 25, CDS at NYU. Brett Bernstein (CDS at NYU) Lecture 13 April 25, / 26 Principal Component Analysis Brett Bernstein CDS at NYU April 25, 2017 Brett Bernstein (CDS at NYU) Lecture 13 April 25, 2017 1 / 26 Initial Question Intro Question Question Let S R n n be symmetric. 1

More information

Lecture 10 - Eigenvalues problem

Lecture 10 - Eigenvalues problem Lecture 10 - Eigenvalues problem Department of Computer Science University of Houston February 28, 2008 1 Lecture 10 - Eigenvalues problem Introduction Eigenvalue problems form an important class of problems

More information

Math 2B Spring 13 Final Exam Name Write all responses on separate paper. Show your work for credit.

Math 2B Spring 13 Final Exam Name Write all responses on separate paper. Show your work for credit. Math 2B Spring 3 Final Exam Name Write all responses on separate paper. Show your work for credit.. True or false, with reason if true and counterexample if false: a. Every invertible matrix can be factored

More information

Dimension reduction, PCA & eigenanalysis Based in part on slides from textbook, slides of Susan Holmes. October 3, Statistics 202: Data Mining

Dimension reduction, PCA & eigenanalysis Based in part on slides from textbook, slides of Susan Holmes. October 3, Statistics 202: Data Mining Dimension reduction, PCA & eigenanalysis Based in part on slides from textbook, slides of Susan Holmes October 3, 2012 1 / 1 Combinations of features Given a data matrix X n p with p fairly large, it can

More information

Econ Slides from Lecture 7

Econ Slides from Lecture 7 Econ 205 Sobel Econ 205 - Slides from Lecture 7 Joel Sobel August 31, 2010 Linear Algebra: Main Theory A linear combination of a collection of vectors {x 1,..., x k } is a vector of the form k λ ix i for

More information

Introduction to Matrix Algebra

Introduction to Matrix Algebra Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p

More information

EE731 Lecture Notes: Matrix Computations for Signal Processing

EE731 Lecture Notes: Matrix Computations for Signal Processing EE731 Lecture Notes: Matrix Computations for Signal Processing James P. Reilly c Department of Electrical and Computer Engineering McMaster University September 22, 2005 0 Preface This collection of ten

More information

3D Computer Vision - WT 2004

3D Computer Vision - WT 2004 3D Computer Vision - WT 2004 Singular Value Decomposition Darko Zikic CAMP - Chair for Computer Aided Medical Procedures November 4, 2004 1 2 3 4 5 Properties For any given matrix A R m n there exists

More information

Information Retrieval

Information Retrieval Introduction to Information CS276: Information and Web Search Christopher Manning and Pandu Nayak Lecture 13: Latent Semantic Indexing Ch. 18 Today s topic Latent Semantic Indexing Term-document matrices

More information

Math for ML: review. CS 1675 Introduction to ML. Administration. Lecture 2. Milos Hauskrecht 5329 Sennott Square, x4-8845

Math for ML: review. CS 1675 Introduction to ML. Administration. Lecture 2. Milos Hauskrecht 5329 Sennott Square, x4-8845 CS 75 Introduction to ML Lecture Math for ML: review Milos Hauskrecht milos@cs.pitt.edu 5 Sennott Square, x4-45 people.cs.pitt.edu/~milos/courses/cs75/ Administration Instructor: Prof. Milos Hauskrecht

More information

Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Exam 2 will be held on Tuesday, April 8, 7-8pm in 117 MacMillan What will be covered The exam will cover material from the lectures

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: Numerical Analysis I (Numerical Linear Algebra) AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 1: Course Overview & Matrix-Vector Multiplication Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 20 Outline 1 Course

More information

Name Solutions Linear Algebra; Test 3. Throughout the test simplify all answers except where stated otherwise.

Name Solutions Linear Algebra; Test 3. Throughout the test simplify all answers except where stated otherwise. Name Solutions Linear Algebra; Test 3 Throughout the test simplify all answers except where stated otherwise. 1) Find the following: (10 points) ( ) Or note that so the rows are linearly independent, so

More information

Eigenvalue and Eigenvector Homework

Eigenvalue and Eigenvector Homework Eigenvalue and Eigenvector Homework Olena Bormashenko November 4, 2 For each of the matrices A below, do the following:. Find the characteristic polynomial of A, and use it to find all the eigenvalues

More information

(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =

(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax = . (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)

More information

Math 315: Linear Algebra Solutions to Assignment 7

Math 315: Linear Algebra Solutions to Assignment 7 Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are

More information

1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued)

1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued) 1 A linear system of equations of the form Sections 75, 78 & 81 a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m can be written in matrix

More information

Pseudoinverse and Adjoint Operators

Pseudoinverse and Adjoint Operators ECE 275AB Lecture 5 Fall 2008 V1.1 c K. Kreutz-Delgado, UC San Diego p. 1/1 Lecture 5 ECE 275A Pseudoinverse and Adjoint Operators ECE 275AB Lecture 5 Fall 2008 V1.1 c K. Kreutz-Delgado, UC San Diego p.

More information

Principal Component Analysis

Principal Component Analysis Principal Component Analysis Yingyu Liang yliang@cs.wisc.edu Computer Sciences Department University of Wisconsin, Madison [based on slides from Nina Balcan] slide 1 Goals for the lecture you should understand

More information

Linear Algebra for Machine Learning. Sargur N. Srihari

Linear Algebra for Machine Learning. Sargur N. Srihari Linear Algebra for Machine Learning Sargur N. srihari@cedar.buffalo.edu 1 Overview Linear Algebra is based on continuous math rather than discrete math Computer scientists have little experience with it

More information

Latent Semantic Models. Reference: Introduction to Information Retrieval by C. Manning, P. Raghavan, H. Schutze

Latent Semantic Models. Reference: Introduction to Information Retrieval by C. Manning, P. Raghavan, H. Schutze Latent Semantic Models Reference: Introduction to Information Retrieval by C. Manning, P. Raghavan, H. Schutze 1 Vector Space Model: Pros Automatic selection of index terms Partial matching of queries

More information

forms Christopher Engström November 14, 2014 MAA704: Matrix factorization and canonical forms Matrix properties Matrix factorization Canonical forms

forms Christopher Engström November 14, 2014 MAA704: Matrix factorization and canonical forms Matrix properties Matrix factorization Canonical forms Christopher Engström November 14, 2014 Hermitian LU QR echelon Contents of todays lecture Some interesting / useful / important of matrices Hermitian LU QR echelon Rewriting a as a product of several matrices.

More information

Lecture 7 Spectral methods

Lecture 7 Spectral methods CSE 291: Unsupervised learning Spring 2008 Lecture 7 Spectral methods 7.1 Linear algebra review 7.1.1 Eigenvalues and eigenvectors Definition 1. A d d matrix M has eigenvalue λ if there is a d-dimensional

More information

Exercise Sheet 1. 1 Probability revision 1: Student-t as an infinite mixture of Gaussians

Exercise Sheet 1. 1 Probability revision 1: Student-t as an infinite mixture of Gaussians Exercise Sheet 1 1 Probability revision 1: Student-t as an infinite mixture of Gaussians Show that an infinite mixture of Gaussian distributions, with Gamma distributions as mixing weights in the following

More information

Linear Algebra Review

Linear Algebra Review Linear Algebra Review ORIE 4741 September 1, 2017 Linear Algebra Review September 1, 2017 1 / 33 Outline 1 Linear Independence and Dependence 2 Matrix Rank 3 Invertible Matrices 4 Norms 5 Projection Matrix

More information

https://goo.gl/kfxweg KYOTO UNIVERSITY Statistical Machine Learning Theory Sparsity Hisashi Kashima kashima@i.kyoto-u.ac.jp DEPARTMENT OF INTELLIGENCE SCIENCE AND TECHNOLOGY 1 KYOTO UNIVERSITY Topics:

More information

Lecture 1: Review of linear algebra

Lecture 1: Review of linear algebra Lecture 1: Review of linear algebra Linear functions and linearization Inverse matrix, least-squares and least-norm solutions Subspaces, basis, and dimension Change of basis and similarity transformations

More information

Linear Algebra Review. Fei-Fei Li

Linear Algebra Review. Fei-Fei Li Linear Algebra Review Fei-Fei Li 1 / 37 Vectors Vectors and matrices are just collections of ordered numbers that represent something: movements in space, scaling factors, pixel brightnesses, etc. A vector

More information

Mathematical Properties of Stiffness Matrices

Mathematical Properties of Stiffness Matrices Mathematical Properties of Stiffness Matrices CEE 4L. Matrix Structural Analysis Department of Civil and Environmental Engineering Duke University Henri P. Gavin Fall, 0 These notes describe some of the

More information

DATA MINING LECTURE 8. Dimensionality Reduction PCA -- SVD

DATA MINING LECTURE 8. Dimensionality Reduction PCA -- SVD DATA MINING LECTURE 8 Dimensionality Reduction PCA -- SVD The curse of dimensionality Real data usually have thousands, or millions of dimensions E.g., web documents, where the dimensionality is the vocabulary

More information

Lecture 3: Review of Linear Algebra

Lecture 3: Review of Linear Algebra ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters, transforms,

More information

Deep Learning Book Notes Chapter 2: Linear Algebra

Deep Learning Book Notes Chapter 2: Linear Algebra Deep Learning Book Notes Chapter 2: Linear Algebra Compiled By: Abhinaba Bala, Dakshit Agrawal, Mohit Jain Section 2.1: Scalars, Vectors, Matrices and Tensors Scalar Single Number Lowercase names in italic

More information

STA141C: Big Data & High Performance Statistical Computing

STA141C: Big Data & High Performance Statistical Computing STA141C: Big Data & High Performance Statistical Computing Lecture 5: Numerical Linear Algebra Cho-Jui Hsieh UC Davis April 20, 2017 Linear Algebra Background Vectors A vector has a direction and a magnitude

More information

Lecture 3: Review of Linear Algebra

Lecture 3: Review of Linear Algebra ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak, scribe: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters,

More information

Linear Algebra in Actuarial Science: Slides to the lecture

Linear Algebra in Actuarial Science: Slides to the lecture Linear Algebra in Actuarial Science: Slides to the lecture Fall Semester 2010/2011 Linear Algebra is a Tool-Box Linear Equation Systems Discretization of differential equations: solving linear equations

More information

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the

More information

Linear Algebra (Review) Volker Tresp 2017

Linear Algebra (Review) Volker Tresp 2017 Linear Algebra (Review) Volker Tresp 2017 1 Vectors k is a scalar (a number) c is a column vector. Thus in two dimensions, c = ( c1 c 2 ) (Advanced: More precisely, a vector is defined in a vector space.

More information