Linear Algebra Primer
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1 Linear Algebra Primer Juan Carlo Nieble and Ranja Krihna Sanford Viion and Learning Lab Anoher, ver in-deph linear algebra review from CS229 i available here: hp://c229.anford.edu/ecion/c229-linalg.pdf And a video dicuion of linear algebra from EE263 i here (lecure 3 and 4): hp://ee.anford.edu/coure/ee263 Sanford Univeri 27-Sep-28
2 Ouline Vecor and marice Baic Mari Operaion Deerminan, norm, race Special Marice Tranformaion Marice Homogeneou coordinae Tranlaion Mari invere Mari rank Eigenvalue and Eigenvecor Mari Calculu 27-Sep-28 Sanford Univeri 2
3 Ouline Vecor and marice Baic Mari Operaion Deerminan, norm, race Special Marice Tranformaion Marice Homogeneou coordinae Tranlaion Mari invere Mari rank Eigenvalue and Eigenvecor Mari Calculu Vecor and marice are ju collecion of ordered number ha repreen omehing: movemen in pace, caling facor, piel brighne, ec. We ll define ome common ue and andard operaion on hem. 27-Sep-28 Sanford Univeri 3
4 Vecor A column vecor where A row vecor where denoe he ranpoe operaion 27-Sep-28 Sanford Univeri 4
5 Vecor We ll defaul o column vecor in hi cla You ll wan o keep rack of he orienaion of our vecor when programming in phon You can ranpoe a vecor V in phon b wriing V.. (Bu in cla maerial, we will alwa ue V T o indicae ranpoe, and we will ue V o mean V prime ) 27-Sep-28 Sanford Univeri 5
6 Vecor have wo main ue Vecor can repreen an offe in 2D or 3D pace. Poin are ju vecor from he origin. Daa (piel, gradien a an image kepoin, ec) can alo be reaed a a vecor. Such vecor don have a geomeric inerpreaion, bu calculaion like diance can ill have value. 27-Sep-28 Sanford Univeri 6
7 Mari A mari i an arra of number wih ize b, i.e. m row and n column. If, we a ha i quare. 27-Sep-28 Sanford Univeri 7 7
8 Image Phon repreen an image a a mari of piel brighnee Noe ha he upper lef corner i [,] (,) 27-Sep-28 Sanford Univeri 8 8
9 Image a boh a mari a well a a vecor 27-Sep-28 Sanford Univeri 9
10 Color Image Gracale image have one number per piel, and are ored a an m n mari. Color image have 3 number per piel red, green, and blue brighnee (RGB) Sored a an m n 3 mari 27-Sep-28 Sanford Univeri
11 Baic Mari Operaion We will dicu: Addiion Scaling Do produc Muliplicaion Tranpoe Invere / peudoinvere Deerminan / race 27-Sep-28 Sanford Univeri
12 Mari Operaion Addiion Can onl add a mari wih maching dimenion, or a calar. Scaling 27-Sep-28 Sanford Univeri 2
13 Vecor Norm More formall, a norm i an funcion ha aifie 4 properie: Non-negaivi: For all Definiene: f() if and onl if. Homogenei: For all Triangle inequali: For all 27-Sep-28 Sanford Univeri 3
14 Mari Operaion Eample Norm General norm: 27-Sep-28 Sanford Univeri 4
15 Mari Operaion Inner produc (do produc) of vecor Mulipl correponding enrie of wo vecor and add up he reul i alo Co( he angle beween and ) 27-Sep-28 Sanford Univeri 5
16 Mari Operaion Inner produc (do produc) of vecor If B i a uni vecor, hen A B give he lengh of A which lie in he direcion of B 27-Sep-28 Sanford Univeri 6
17 Mari Operaion The produc of wo marice 27-Sep-28 Sanford Univeri 7
18 Mari Operaion Muliplicaion The produc AB i: Each enr in he reul i (ha row of A) do produc wih (ha column of B) Man ue, which will be covered laer Sanford Univeri 27-Sep-28 8
19 Mari Operaion Muliplicaion eample: Each enr of he mari produc i made b aking he do produc of he correponding row in he lef mari, wih he correponding column in he righ one. 27-Sep-28 Sanford Univeri 9
20 Mari Operaion The produc of wo marice 27-Sep-28 Sanford Univeri 2
21 Mari Operaion Power B convenion, we can refer o he mari produc AA a A 2, and AAA a A 3, ec. Obvioul onl quare marice can be muliplied ha wa 27-Sep-28 Sanford Univeri 2
22 Mari Operaion Tranpoe flip mari, o row become column A ueful ideni: 27-Sep-28 Sanford Univeri 22
23 Mari Operaion Deerminan reurn a calar Repreen area (or volume) of he parallelogram decribed b he vecor in he row of he mari For, Properie: 27-Sep-28 Sanford Univeri 23
24 Mari Operaion Trace Invarian o a lo of ranformaion, o i ued omeime in proof. (Rarel in hi cla hough.) Properie: 27-Sep-28 Sanford Univeri 24
25 Mari Operaion Vecor Norm Mari norm: Norm can alo be defined for marice, uch a he Frobeniu norm: 27-Sep-28 Sanford Univeri 25
26 Special Marice Ideni mari I Square mari, along diagonal, elewhere I [anoher mari] [ha mari] Diagonal mari Square mari wih number along diagonal, elewhere A diagonal [anoher mari] cale he row of ha mari 27-Sep-28 Sanford Univeri 26
27 Special Marice Smmeric mari Skew-mmeric mari Sep-28 Sanford Univeri 27
28 Ouline Vecor and marice Baic Mari Operaion Deerminan, norm, race Special Marice Tranformaion Marice Homogeneou coordinae Tranlaion Mari invere Mari rank Eigenvalue and Eigenvecor Mari Calculu Mari muliplicaion can be ued o ranform vecor. A mari ued in hi wa i called a ranformaion mari. 27-Sep-28 Sanford Univeri 28
29 Tranformaion Marice can be ued o ranform vecor in ueful wa, hrough muliplicaion: A Simple i caling: (Verif o ourelf ha he mari muliplicaion work ou hi wa) 27-Sep-28 Sanford Univeri 29
30 Tranformaion 27-Sep-28 Sanford Univeri 3
31 Roaion! Sep-28 Sanford Univeri 3
32 Roaion How can ou conver a vecor repreened in frame o a new, roaed coordinae frame? 27-Sep-28 Sanford Univeri 32
33 Roaion How can ou conver a vecor repreened in frame o a new, roaed coordinae frame? Remember wha a vecor i: [componen in direcion of he frame ai, componen in direcion of ai] 27-Sep-28 Sanford Univeri 33
34 Roaion So o roae i we mu produce hi vecor: [componen in direcion of new ai, componen in direcion of new ai] We can do hi eail wih do produc! New coordinae i [original vecor] do [he new ai] New coordinae i [original vecor] do [he new ai] 27-Sep-28 Sanford Univeri 34
35 Roaion Inigh: hi i wha happen in a mari*vecor muliplicaion Reul coordinae i: [original vecor] do [mari row ] So mari muliplicaion can roae a vecor p: 27-Sep-28 Sanford Univeri 35
36 Roaion Suppoe we epre a poin in he new coordinae em which i roaed lef If we plo he reul in he original coordinae em, we have roaed he poin righ Thu, roaion marice can be ued o roae vecor. We ll uuall hink of hem in ha ene-- a operaor o roae vecor 27-Sep-28 Sanford Univeri 36
37 2D Roaion Mari Formula Couner-clockwie roaion b an angle q P q P coθ -in θ coθ + in θ coq - in q in q coq 27-Sep-28 Sanford Univeri P R P 37
38 Tranformaion Marice Muliple ranformaion marice can be ued o ranform a poin: p R 2 R S p The effec of hi i o appl heir ranformaion one afer he oher, from righ o lef. In he eample above, he reul i (R 2 (R (S p))) The reul i eacl he ame if we mulipl he marice fir, o form a ingle ranformaion mari: p (R 2 R S) p 27-Sep-28 Sanford Univeri 38
39 Homogeneou em In general, a mari muliplicaion le u linearl combine componen of a vecor Thi i ufficien for cale, roae, kew ranformaion. Bu noice, we can add a conan! L 27-Sep-28 Sanford Univeri 39
40 Homogeneou em The (omewha hack) oluion? Sick a a he end of ever vecor: Now we can roae, cale, and kew like before, AND ranlae (noe how he muliplicaion work ou, above) Thi i called homogeneou coordinae 27-Sep-28 Sanford Univeri 4
41 Homogeneou em In homogeneou coordinae, he muliplicaion work ou o he righmo column of he mari i a vecor ha ge added. Generall, a homogeneou ranformaion mari will have a boom row of [ ], o ha he reul ha a a he boom oo. 27-Sep-28 Sanford Univeri 4
42 Homogeneou em One more hing we migh wan: o divide he reul b omehing For eample, we ma wan o divide b a coordinae, o make hing cale down a he ge farher awa in a camera image Mari muliplicaion can acuall divide So, b convenion, in homogeneou coordinae, we ll divide he reul b i la coordinae afer doing a mari muliplicaion 27-Sep-28 Sanford Univeri 42
43 2D Tranlaion P P 27-Sep-28 Sanford Univeri 43
44 Sanford Univeri 27-Sep P P + + P,), ( ), (,), ( ), ( P P 2D Tranlaion uing Homogeneou Coordinae
45 Sanford Univeri 27-Sep P P + + P,), ( ), (,), ( ), ( P P 2D Tranlaion uing Homogeneou Coordinae
46 Sanford Univeri 27-Sep P P + + P,), ( ), (,), ( ), ( P P 2D Tranlaion uing Homogeneou Coordinae
47 Sanford Univeri 27-Sep P P + + P,), ( ), (,), ( ), ( P P 2D Tranlaion uing Homogeneou Coordinae
48 Sanford Univeri 27-Sep P P + + P,), ( ), (,), ( ), ( P P P T P I 2D Tranlaion uing Homogeneou Coordinae
49 Scaling P P 27-Sep-28 Sanford Univeri 49
50 Scaling Equaion P P P (, ) P (, ) P (, ) (,,) P (, ) (,,) 27-Sep-28 Sanford Univeri 5
51 Sanford Univeri 27-Sep-28 5 Scaling Equaion P P P,), ( ), (,), ( ), ( P P ), ( ), ( P P
52 Sanford Univeri 27-Sep Scaling Equaion P P P,), ( ), (,), ( ), ( P P S P S P S ), ( ), ( P P
53 Scaling & Tranlaing P P P S P P T P P T P T (S P) T S P 27-Sep-28 Sanford Univeri 53
54 Sanford Univeri 27-Sep Scaling & Tranlaing P T S P " # $ $ $ $ % & " # $ $ $ $ % & " # $ $ $ % & " # $ $ $ $ % & " # $ $ $ % & + + " # $ $ $ $ % & S " # $ % & " # $ $ $ % & A
55 Sanford Univeri 27-Sep Scaling & Tranlaing P T S P " # $ $ $ $ % & " # $ $ $ $ % & " # $ $ $ % & " # $ $ $ $ % & " # $ $ $ % & + + " # $ $ $ $ % & S " # $ % & " # $ $ $ % &
56 Sanford Univeri 27-Sep P S T P Tranlaing & Scaling! Scaling & Tranlaing
57 Sanford Univeri 27-Sep P T S P + + P S T P Tranlaing & Scaling! Scaling & Tranlaing
58 Sanford Univeri 27-Sep Tranlaing & Scaling! Scaling & Tranlaing + + P T S P + + P S T P
59 Roaion P P 27-Sep-28 Sanford Univeri 59
60 Roaion Equaion Couner-clockwie roaion b an angle q P q P coθ -in θ coθ + in θ coq - in q in q coq 27-Sep-28 Sanford Univeri P R P 6
61 Roaion Mari Properie coq - in q in q coq A 2D roaion mari i 22 Noe: R belong o he caegor of normal marice and aifie man inereing properie: Sanford Univeri R R T R de( R) T R I 27-Sep-28 6
62 Roaion Mari Properie Tranpoe of a roaion mari produce a roaion in he oppoie direcion R R T de( R) R The row of a roaion mari are alwa muuall perpendicular (a.k.a. orhogonal) uni vecor (and o are i column) R T I 27-Sep-28 Sanford Univeri 62
63 Sanford Univeri 27-Sep Scaling + Roaion + Tranlaion P (T R S) P - coθ inθ in θ coθ R P S T P - coθ inθ in θ coθ R S S R Thi i he form of he general-purpoe ranformaion mari
64 Ouline Vecor and marice Baic Mari Operaion Deerminan, norm, race Special Marice Tranformaion Marice Homogeneou coordinae Tranlaion Mari invere Mari rank Eigenvalue and Eigenvecor Mari Calculae The invere of a ranformaion mari revere i effec 27-Sep-28 Sanford Univeri 64
65 Invere Given a mari A, i invere A - i a mari uch ha AA - A - A I E.g. Invere doe no alwa ei. If A - ei, A i inverible or non-ingular. Oherwie, i ingular. Ueful ideniie, for marice ha are inverible: 27-Sep-28 Sanford Univeri 65
66 Mari Operaion Peudoinvere Forunael, here are workaround o olve AXB in hee iuaion. And phon can do hem! Inead of aking an invere, direcl ak phon o olve for X in AXB, b ping np.linalg.olve(a, B) Phon will r everal appropriae numerical mehod (including he peudoinvere if he invere doen ei) Phon will reurn he value of X which olve he equaion If here i no eac oluion, i will reurn he cloe one If here are man oluion, i will reurn he malle one 27-Sep-28 Sanford Univeri 66
67 Mari Operaion Phon eample: >> impor nump a np >> np.linalg.olve(a,b). -.5 Sanford Univeri 27-Sep-28 67
68 Ouline Vecor and marice Baic Mari Operaion Deerminan, norm, race Special Marice Tranformaion Marice Homogeneou coordinae Tranlaion Mari invere Mari rank Eigenvalue and Eigenvecor Mari Calculae The rank of a ranformaion mari ell ou how man dimenion i ranform a vecor o. 27-Sep-28 Sanford Univeri 68
69 Linear independence Suppoe we have a e of vecor v,, v n If we can epre v a a linear combinaion of he oher vecor v 2 v n, hen v i linearl dependen on he oher vecor. The direcion v can be epreed a a combinaion of he direcion v 2 v n. (E.g. v.7 v v 4 ) 27-Sep-28 Sanford Univeri 69
70 Linear independence Suppoe we have a e of vecor v,, v n If we can epre v a a linear combinaion of he oher vecor v 2 v n, hen v i linearl dependen on he oher vecor. The direcion v can be epreed a a combinaion of he direcion v 2 v n. (E.g. v.7 v v 4 ) If no vecor i linearl dependen on he re of he e, he e i linearl independen. Common cae: a e of vecor v,, v n i alwa linearl independen if each vecor i perpendicular o ever oher vecor (and non-zero) 27-Sep-28 Sanford Univeri 7
71 Linear independence Linearl independen e No linearl independen 27-Sep-28 Sanford Univeri 7
72 Mari rank Column/row rank Column rank alwa equal row rank Mari rank 27-Sep-28 Sanford Univeri 72
73 Mari rank For ranformaion marice, he rank ell ou he dimenion of he oupu E.g. if rank of A i, hen he ranformaion p Ap map poin ono a line. Here a mari wih rank : All poin ge mapped o he line 2 27-Sep-28 Sanford Univeri 73
74 Mari rank If an m m mari i rank m, we a i full rank Map an m vecor uniquel o anoher m vecor An invere mari can be found If rank < m, we a i ingular A lea one dimenion i geing collaped. No wa o look a he reul and ell wha he inpu wa Invere doe no ei Invere alo doen ei for non-quare marice 27-Sep-28 Sanford Univeri 74
75 Ouline Vecor and marice Baic Mari Operaion Deerminan, norm, race Special Marice Tranformaion Marice Homogeneou coordinae Tranlaion Mari invere Mari rank Eigenvalue and Eigenvecor(SVD) Mari Calculu 27-Sep-28 Sanford Univeri 75
76 Eigenvecor and Eigenvalue An eigenvecor of a linear ranformaion A i a non-zero vecor ha, when A i applied o i, doe no change direcion. 27-Sep-28 Sanford Univeri 76
77 Eigenvecor and Eigenvalue An eigenvecor of a linear ranformaion A i a non-zero vecor ha, when A i applied o i, doe no change direcion. Appling A o he eigenvecor onl cale he eigenvecor b he calar value λ, called an eigenvalue. 27-Sep-28 Sanford Univeri 77
78 Eigenvecor and Eigenvalue We wan o find all he eigenvalue of A: Which can we wrien a: Therefore: 27-Sep-28 Sanford Univeri 78
79 Eigenvecor and Eigenvalue We can olve for eigenvalue b olving: Since we are looking for non-zero, we can inead olve he above equaion a: 27-Sep-28 Sanford Univeri 79
80 Properie The race of a A i equal o he um of i eigenvalue: The deerminan of A i equal o he produc of i eigenvalue The rank of A i equal o he number of non-zero eigenvalue of A. The eigenvalue of a diagonal mari D diag(d,... dn) are ju he diagonal enrie d,... dn Sanford Univeri 27-Sep-28 8
81 Specral heor We call an eigenvalue λ and an aociaed eigenvecor an eigenpair. The pace of vecor where (A λi) i ofen called he eigenpace of A aociaed wih he eigenvalue λ. The e of all eigenvalue of A i called i pecrum: 27-Sep-28 Sanford Univeri 8
82 Specral heor The magniude of he large eigenvalue (in magniude) i called he pecral radiu Where C i he pace of all eigenvalue of A 27-Sep-28 Sanford Univeri 82
83 Specral heor The pecral radiu i bounded b infini norm of a mari: Proof: Turn o a parner and prove hi! 27-Sep-28 Sanford Univeri 83
84 Specral heor The pecral radiu i bounded b infini norm of a mari: Proof: Le λ and v be an eigenpair of A: 27-Sep-28 Sanford Univeri 84
85 Diagonalizaion An n n mari A i diagonalizable if i ha n linearl independen eigenvecor. Mo quare marice (in a ene ha can be made mahemaicall rigorou) are diagonalizable: Normal marice are diagonalizable Marice wih n diinc eigenvalue are diagonalizable Lemma: Eigenvecor aociaed wih diinc eigenvalue are linearl independen. 27-Sep-28 Sanford Univeri 85
86 Diagonalizaion An n n mari A i diagonalizable if i ha n linearl independen eigenvecor. Mo quare marice are diagonalizable: Normal marice are diagonalizable Marice wih n diinc eigenvalue are diagonalizable Lemma: Eigenvecor aociaed wih diinc eigenvalue are linearl independen. 27-Sep-28 Sanford Univeri 86
87 Diagonalizaion Eigenvalue equaion: Where D i a diagonal mari of he eigenvalue 27-Sep-28 Sanford Univeri 87
88 Diagonalizaion Eigenvalue equaion: Auming all λ i are unique: Remember ha he invere of an orhogonal mari i ju i ranpoe and he eigenvecor are orhogonal 27-Sep-28 Sanford Univeri 88
89 Smmeric marice Properie: For a mmeric mari A, all he eigenvalue are real. The eigenvecor of A are orhonormal. 27-Sep-28 Sanford Univeri 89
90 Smmeric marice Therefore: where So, wha can ou a abou he vecor ha aifie he following opimizaion? 27-Sep-28 Sanford Univeri 9
91 Smmeric marice Therefore: where So, wha can ou a abou he vecor ha aifie he following opimizaion? I he ame a finding he eigenvecor ha correpond o he large eigenvalue of A. 27-Sep-28 Sanford Univeri 9
92 Some applicaion of Eigenvalue PageRank Schrodinger equaion PCA We are going o ue i o compre image in fuure clae 27-Sep-28 Sanford Univeri 92
93 Ouline Vecor and marice Baic Mari Operaion Deerminan, norm, race Special Marice Tranformaion Marice Homogeneou coordinae Tranlaion Mari invere Mari rank Eigenvalue and Eigenvecor(SVD) Mari Calculu 27-Sep-28 Sanford Univeri 93
94 Mari Calculu The Gradien Le a funcion ake a inpu a mari A of ize m n and reurn a real value. Then he gradien of f: 27-Sep-28 Sanford Univeri 94
95 Mari Calculu The Gradien Ever enr in he mari i: he ize of A f(a) i alwa he ame a he ize of A. So if A i ju a vecor : 27-Sep-28 Sanford Univeri 95
96 Eercie Eample: Find: 27-Sep-28 Sanford Univeri 96
97 Eercie Eample: From hi we can conclude ha: 27-Sep-28 Sanford Univeri 97
98 Mari Calculu The Gradien Properie 27-Sep-28 Sanford Univeri 98
99 Mari Calculu The Heian The Heian mari wih repec o, wrien impl a H: or The Heian of n-dimenional vecor i he n n mari. 27-Sep-28 Sanford Univeri 99
100 Mari Calculu The Heian Each enr can be wrien a: Eercie: Wh i he Heian alwa mmeric? 27-Sep-28 Sanford Univeri
101 Mari Calculu The Heian Each enr can be wrien a: The Heian i alwa mmeric, becaue Thi i known a Schwarz heorem: The order of parial derivaive don maer a long a he econd derivaive ei and i coninuou. 27-Sep-28 Sanford Univeri
102 Mari Calculu The Heian Noe ha he heian i no he gradien of whole gradien of a vecor (hi i no defined). I i acuall he gradien of ever enr of he gradien of he vecor. 27-Sep-28 Sanford Univeri 2
103 Mari Calculu The Heian Eg, he fir column i he gradien of 27-Sep-28 Sanford Univeri 3
104 Eercie Eample: 27-Sep-28 Sanford Univeri 4
105 Eercie 27-Sep-28 Sanford Univeri 5
106 Eercie Divide he ummaion ino 3 par depending on wheher: i k or j k 27-Sep-28 Sanford Univeri 6
107 Eercie 27-Sep-28 Sanford Univeri 7
108 Eercie 27-Sep-28 Sanford Univeri 8
109 Eercie 27-Sep-28 Sanford Univeri 9
110 Eercie 27-Sep-28 Sanford Univeri
111 Eercie 27-Sep-28 Sanford Univeri
112 Eercie 27-Sep-28 Sanford Univeri 2
113 Eercie 27-Sep-28 Sanford Univeri 3
114 Eercie 27-Sep-28 Sanford Univeri 4
115 Eercie 27-Sep-28 Sanford Univeri 5
116 Wha we have learned Vecor and marice Baic Mari Operaion Special Marice Tranformaion Marice Homogeneou coordinae Tranlaion Mari invere Mari rank Eigenvalue and Eigenvecor Mari Calculae 27-Sep-28 Sanford Univeri 6
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