Matrix Calculations: Applications of Eigenvalues and Eigenvectors; Inner Products
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1 Matrix Calculations: Applications of Eigenvalues and Eigenvectors; Inner Products H. Geuvers Institute for Computing and Information Sciences Intelligent Systems Version: spring 2015 H. Geuvers Version: spring 2015 Matrix Calculations 1 / 36
2 Outline Applications of Eigenvalues and Eigenvectors Applications of Eigenvalues and Eigenvectors H. Geuvers Version: spring 2015 Matrix Calculations 2 / 36
3 Political swingers re-re-revisited, part I Recall the political transisition matrix ( ) P = = 1 10 ( ) Eigenvalues λ are obtained via det(p λ I 2 ) = 0: ( 8 10 λ)( 9 10 λ) = λ λ = 0 Solutions via abc ( 1 17 ( 2 10 ± 17 ) ) = 2( ± = 2( ± ) = 2( ± 3 10 Hence λ = = 1 or λ = = H. Geuvers Version: spring 2015 Matrix Calculations 4 / 36 ) )
4 Political swingers re-re-revisited, part II λ = 1 Indeed solve: { 0.2x + 0.1y = 0 0.2x + 0.1y = 0 ) ( ) 1 = 2 ( ( giving (1, 2) as eigenvector ) = ( ) 1 2 = 1 ( ) 1 2 λ = 0.7 solve: Check: ( ) { 0.1x + 0.1y = 0 0.2x + 0.2y = 0 ( ) 1 1 = ( ) giving (1, 1) as eigenvector = ( ) ( ) 1 = H. Geuvers Version: spring 2015 Matrix Calculations 5 / 36
5 Political swingers re-re-revisited, part III The eigenvalues 1 and 0.7 are different, and indeed the eigenvectors (1, 2) and (1, 1) are independent The coordinate-translation T V B from the eigenvector basis V = {(1, 2), (1, 1)} to the standard basis B = {(1, 0), (0, 1)} consists of the eigenvectors: ( ) 1 1 T V B = 2 1 In the reverse direction: T B V = ( T V B ) 1 = ( ) = 1 3 ( ) H. Geuvers Version: spring 2015 Matrix Calculations 6 / 36
6 Political swingers re-re-revisited, part IV We explicitly check the diagonalisation equation: ( ) ( ) ( ) ( ) T V B T B V = ( ) ( ) = ( ) = 1 3 = ( ) = P, the original political transition matrix! H. Geuvers Version: spring 2015 Matrix Calculations 7 / 36
7 Political swingers re-re-revisited, part V ( ) 1 0 This diagonalisation P = T T is useful for iteration ( ) ( ) P 2 = T T T T ( ) ( ) = T T ( ) = T 0 (0.7) 2 T 1 ( ) (1) P n n 0 = T 0 (0.7) n T 1 ( ) 1 0 lim n Pn = T T since lim n (0.7)n = 0 ( ) ( ) ( ) ( ) = = H. Geuvers Version: spring 2015 Matrix Calculations 8 / 36
8 Political swingers re-re-revisited, part VI In an earlier lecture we wondered how to compute P n We can now see that in the limit it goes to: ( ) ( ) ( ) 100 lim n Pn = ( 2 2 ) 150 ( ) = = ( ) (This was already suggested earlier, but now we can calculate it!) Recall the useful limit result lim n an = 0, for a < 1. H. Geuvers Version: spring 2015 Matrix Calculations 9 / 36
9 Rental car returns, part I Assume a car rental company with three locations, for picking up and returning cars, written as P, Q, R The weekly distribution history shows: Location P 60% stay at P 10% go to Q 30% go to R Location Q 10% go to P 80% stay at Q 10% go to R Location R 10% go to P 20% go to Q 70% stay at R H. Geuvers Version: spring 2015 Matrix Calculations 10 / 36
10 Applications of Eigenvalues and Eigenvectors Rental car returns, part II Two possible representations of these return distributions 1 As probabilistic transition system P Q R H. Geuvers Version: spring 2015 Matrix Calculations 11 / 36
11 Rental car returns, part III 2 As a transition matrix C = = This matrix C describes what is called a Markov chain: all entries are in the unit interval [0, 1] of probabilities in each column, the entries add up to 1 H. Geuvers Version: spring 2015 Matrix Calculations 12 / 36
12 Rental car returns, part IV Task: Start from the following division of cars: P 200 P = Q = R = 200 ie. Q = 200 R 200 Determine the division of cars after two weeks Determine the equilibrium division, reached as the number of weeks goes to infinity H. Geuvers Version: spring 2015 Matrix Calculations 13 / 36
13 Rental car returns, part V After one week we have: C 200 = = = After two weeks we have: C 220 = = = H. Geuvers Version: spring 2015 Matrix Calculations 14 / 36
14 Rental car returns, part VI For the equilibrium we first compute eigenvalues and eigenvectors of the transition matrix C The characteristic polynomial is: 0.6 λ λ λ = λ λ λ [ ( ) = (6 10λ) (8 10λ)(7 10λ) 2 ( ) ( )] 1 (7 10λ) (8 10λ) = = [ ] 1000λ λ λ = λ λ 2 1.4λ H. Geuvers Version: spring 2015 Matrix Calculations 15 / 36
15 Rental car returns, part VII Next we solve λ λ 2 1.4λ = 0. We seek a trivial solution; again λ = 1 works! Now we can write λ λ 2 1.4λ = (λ 1)( λ λ 0.3) We can apply the abc formula to the second part: 1.1± (1.1) = 1.1± = 1.1± = 1.1±0.1 2 This yields additional eigenvalues: λ = 0.5 and λ = 0.6. H. Geuvers Version: spring 2015 Matrix Calculations 16 / 36
16 Rental car returns, part VIII λ = 1 has eigenvector (4, 9, 7); indeed: C 9 = = = λ = 0.6 has eigenvector (0, 1, 1): C 1 = = = λ = 0.5 has eigenvector ( 1, 1, 2): C 1 = = = H. Geuvers Version: spring 2015 Matrix Calculations 17 / 36
17 Rental car returns, part IX Now: eigenvector base V = {(4, 9, 7), (0, 1, 1), ( 1, 1, 2)} and standard base as B = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}. Then we can do change-of-coordinates back-and-forth: T V B = T B V = These translation matrices yield a diagonalisation: C = T V B T B V H. Geuvers Version: spring 2015 Matrix Calculations 18 / 36
18 Rental car returns, part X Thus: 1 n 0 0 lim C n = lim T V B 0 (0.6) n 0 T B V n n 0 0 (0.5) n = T V B T B V = Finally, the equilibrium starting from P = Q = R = 200 is: = H. Geuvers Version: spring 2015 Matrix Calculations 19 / 36
19 Length of a vector Intuitively, each vector v = (x 1,..., x n ) R n has a length (aka. size or norm or Euclidian length), written as v This v is a non-negative real number: v R, v 0 Some special cases: n = 1: so v R, with v = v n = 2: so v = (x 1, x 2 ) R 2 and with Pythagoras: v 2 = x1 2 + x2 2 and thus v = x1 2 + x 2 2 n = 3: so v = (x 1, x 2, x 3 ) R 3 and also with Pythagoras: v 2 = x1 2 + x2 2 + x3 2 and thus v = x1 2 + x x 3 2 In general, for v = (x 1,..., x n ) R n, v = x1 2 + x x n 2 H. Geuvers Version: spring 2015 Matrix Calculations 21 / 36
20 Distance between points Assume now we have two vectors v, w R n, written as: v = (x 1,..., x n ) w = (y 1,..., y n ) What is the distance between the endpoints? commonly written as d(v, w) again, d(v, w) is a non-negative real For n = 2, d(v, w) = (x 1 y 1 ) 2 + (x 2 y 2 ) 2 = v w = w v This will be used also for other n, so: d(v, w) = v w H. Geuvers Version: spring 2015 Matrix Calculations 22 / 36
21 Length is fundamental Distance can be obtained from length of vectors Interestingly, also angles can be obtained from length! Both length of vectors and angles between vectors can de derived from the notion of inner product H. Geuvers Version: spring 2015 Matrix Calculations 23 / 36
22 Inner product definition Definition For vectors v = (x 1,..., x n ), w = (y 1,..., y n ) R n define their inner product as the real number: v, w = x 1 y x n y n = x i y i 1 i n Note: Length v can be expressed via inner product: v 2 = x x 2 n = v, v, so v = v, v. H. Geuvers Version: spring 2015 Matrix Calculations 24 / 36
23 via matrix transpose Recall matrix transposition For an m n matrix A, the transpose A T is the n m matrix A obtained by mirroring in the diagonal: T a 11 a 1n a 11 a m1. =. a m1 a mn a 1n a mn The inner product of v = (x 1,..., x n ), w = (y 1,..., y n ) R n is then a matrix product: v, w = x 1 y x n y n = (x 1 x n ). = v T w. y 1 y n H. Geuvers Version: spring 2015 Matrix Calculations 25 / 36
24 Applications of Eigenvalues and Eigenvectors and angles, part I 0 (1,0),(2,0) =2 0 (1,0),(1,1) =1 0 (1,0),(0,1) =0 0 0 (1,0),( 1,1) = 1 0 (1,0),( 1,0) = 1 (1,0),( 1, 1) = (1,0),(0, 1) =0 (1,0),(1, 1) =1 H. Geuvers Version: spring 2015 Matrix Calculations 26 / 36
25 Reminder: cosine law C γ b d A a D c 2 = a 2 + b 2 2ab cos(γ) h c B Proof: By Pythagoras b 2 = h 2 + d 2 and c 2 = h 2 + (a d) 2. Hence by subtraction: b 2 c 2 = d 2 a 2 +2ad d 2 = a 2 +2ad and so c 2 = a 2 +b 2 2ad Recall cos(γ) = d b, so by substituting d = b cos(γ) we are done. H. Geuvers Version: spring 2015 Matrix Calculations 27 / 36
26 and angles, part II w w d(v, w) = v w 0 γ v v The cosine rule says: That is: v w 2 Let s elaborate it... = v 2 + w 2 2 v w cos(γ) cos(γ) = v 2 + w 2 v w 2 2 v w H. Geuvers Version: spring 2015 Matrix Calculations 28 / 36
27 and angles, part III Starting from the cosine rule: cos(γ) = v 2 + w 2 v w 2 2 v w = x x 2 n + y y 2 n (x 1 y 1 ) 2 (x n y n ) 2 2 v w = 2x 1y x n y n 2 v w = x 1y x n y n v w v, w = v w remember this: cos(γ) = v, w v w Thus, angles between vectors are expressible via the inner product (since v = v, v ). H. Geuvers Version: spring 2015 Matrix Calculations 29 / 36
28 Recall the cosine function H. Geuvers Version: spring 2015 Matrix Calculations 30 / 36
29 Linear algebra in gaming, part I Linear algebra plays an important role in game visualisation Here: simple illustration, borrowed from blog.wolfire.com (More precisely: linear-algebra-for-game-developers-part-2) Recall: cosine cos function is positive on angles between -90 and +90 degrees. H. Geuvers Version: spring 2015 Matrix Calculations 31 / 36
30 Linear algebra in gaming, part II Consider a guard G and hero H in: The guard is at position (1, 1), facing in direction D = with a 180 degrees field of view The hero is at (3, 0). Is he within view? ( ) 1, 1 H. Geuvers Version: spring 2015 Matrix Calculations 32 / 36
31 Linear algebra in gaming, part III The direction vector V is: V = ( ) 3 0 ( ) 1 = 1 ( ) 2 1 The angle γ between D and V must be between -90 and +90! Hence we must have: cos(γ) = D,V D V 0 Since D 0 and V 0, it suffices to have: D, V 0 Well, D, V = = 1. Hence H is within sight! H. Geuvers Version: spring 2015 Matrix Calculations 33 / 36
32 Linear algebra in gaming, part IV Now what if the guard s field of view is 60 degrees? Inbetween -30 and +30 degrees we have cos The cosine of the actual angle γ between D and V is: cos(γ) = D, V D V = ( 1) 2 1 = 0.31 < H is now out of view! (the angle γ = cos 1 (0.31) = 72 degr.) H. Geuvers Version: spring 2015 Matrix Calculations 34 / 36
33 Inner product for vector spaces in general Definition We say that a vector space V has an inner product if there is a special function:, V V R satisfying the following five requirements. 1 v, v 0 2 v, v = 0 if and only if v = 0 3 v, w = w, v 4 v + v, w = v, w + v, w (similarly in w, by 3) 5 av, w = a v, w (and similarly in w, by 3) Given such inner product, define length, distance and angle: v = v, w v, v d(v, w) = v w cos(γ) = v w. H. Geuvers Version: spring 2015 Matrix Calculations 35 / 36
34 Hilbert spaces The notion of inner product turns out to be very general and flexible It combines algebra (vectors) and geometry (distance) It forms the basis of Hilbert spaces (involving completeness ) Our examples: inner product on R n Many other examples exist, involving for instance distance between functions Important topic in abstract analysis and (quantum mechanics) Our main applications: projections and approximations H. Geuvers Version: spring 2015 Matrix Calculations 36 / 36
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