CSE590B Lecture 4 More about P 1

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1 SE590 Lece 4 Moe abo P 1 Tansfoming Tansfomaions James. linn Jimlinn.om h://coses.cs.washingon.ed/coses/cse590b/13a/

2 Peviosly On SE590b

3 Tansfomaions M M w w w w w

4 The ncion w w w w w w 0 w w 0 w 0 w

5 The Phase Sace of M M M M M M M d M M M M M M 0 M M M M 0 Possible nmeic signaes singla d lso a valid mai Niloen d an d, 1

6 New oodinae Sysem E G H G H E E E G H G H E H G E 4 d 4 G H E H G E d H G

7 Plo in EGH sace H G 0 1 E E E H/E H G d 0 E E E G/E 0 lane a infiniy omae wih Q vesion /E /E /E

8 Roadma of M oaions singla ideniy Niloen d = 0 Single eigenvale Involions (ace=0)

9 nd Now

10 inding Real Eigenvales ind sch ha de MI 0 Linea combo of M and I is singla I singla M Oside cone (ed,ble): Two inesecions on line M Inside cone (geen): No inesecions on line

11 Tansfomaion of Tansfomaions * T T v s v s v v s s sv vv ss sv s v s v s s s v s v s v v v vv

12 Tansfomaion of Tansfomaions s s s v s v s v v v vv E G H G H E E G H E v s E 0 v s s v s v 1 1 G 0 sv ss vv ss vvg 1 1 H 0 sv ss vv ss vvh

13 isill wih Roaion Tansfom E v s E v s s v s v sv ss vv ss vv G sv ss vv ss vv H cos sin 0 s v sin cos 1 1 G H 0 E E 0 cos sin 0 G 0 sin cos 0 G H H Simila o wha we did wih Q: E E

14 Roaion Tansfom Roae o make G zeo E E 0 cos sin sin cos 0 G H H Noe: No dividing by E ye E H 1 4 d H H H = 0 0 d = 0 1 E ~ H d = 0 ~ E

15 Roaion Tansfom Roae o make G zeo E E 0 cos sin sin cos 0 G H H Pojec ono ni shee E H 1 4 d H H H 1 E H d E

16 Maings Qadaic Polys (3) /E oae E ojec /E E Tansfomaion m (4) H oae [E,,G,H] ojec E

17 Sign lis Roaing 90 degees flis (,G) sign E E G G H H Scaling by -1 in flis (E,) signs E E G G H H Echanging,w flis (E,G) signs d E H E E G G H H

18 Sign lis H Echanging,w flis (E,G) signs E E G G H H d E

19 Sign li effec on angle ange +180 o -180 o +90 o -90 o 0 o E H +170 o -90 o -10 o 0 o E +90 o H +10 o 10 /

20 he ansfom ha kees G=0 d E H E v s 0 0 E 0 v s s v 1 G 0 sv ss vv H 1 H 0 sv ss vv sv 0 ss vv 0 s s o v v = iagonal scale

21 Effec of iagonal Scale E 0 0 E 0 H 0 H d 4 4E H E E H H H E E H H d d E 0 d E H

22 Effec of iagonal Scale =0 0 d de H H 0 0 d E lane H 0 0 d E H cone along ais 0 0 inesecing lanes 0 0 d E H cone along H ais neg 0 0 d E lane neg singla d 0 E Niloen d

23 iagonal Scale o ge o H zeo E 0 0 E 0 H 0 H H, discim H H H H, discim H If osiive can make =0 If osiive can make H=0 H H

24 d=0 case E 0 0 E 0 H 0 H H E 0 E 0 0 E 0 H E 0 E 0 0 E 0 H H

25 isilled EGH Sace H E E E H H E

26 Phase gm and isilled M M 0 M M M M 0 E E Niloen singla d H M M M M M M E H H E d M M M M

27 Phase gm and isilled Q M M 0 M M M M 0 E E I E Niloen singla M M M M M M d isingish beween V R H d M M M M M V R H V R M V R M M V R

28 Phase gm and isilled M M 0 M M M M 0 E E Niloen singla d H M M M M M M d M M M M M V R H Use if d 0

29 Inenal Sce

30 Using oe odcs o make M M = k n

31 Wie k,n in ems of, M = k n k = a + b n = c + d M = a b c d

32 Tansfomaion T* M T = a T* T b T* c T* T d T* T T T* T T ~ T = ~ T = ~ ~ M = a ~ ~ b ~ ~ c ~ ~ d ~ ~

33 Pick nice, = 0 1 = M ~ ~ ~ + ~ ~ ~ ~ ~ ~

34 Niloen E E H isilled: E G H 1 1 G H E 1 1 E 0 1 G 0 H 1 an Tansfom o: E G H 0 1 G H E 0 0 E 0 0 G 1 H 1

35 Niloen N N N w w a N a N H d

36 Idemoen H a a 0 w w d

37 n Ideniy H/E ^ I G/E /E

38 Geneal Scales (Eigenvecos) w E 0 E 0 w M = (E+) +(E) M = (E+) w M = + (-E) w E/H G/H H/E G/E H /H /E d

39 Scale Involion (Eigenvecos) w E 0 0, E 0 E w V S = + M = M = E/H H/E G/H G/E H /E /H d

40 Roaion w E H H E E w + H E ^I + H H d

41 Roaion Involion E H, E 0 H E + H w w H d

42 Single eigenvale d = 0 w w ^I H d

43 Pe & Mied Tensos - Relaion Q T /E /E Q T =0 1,w a,b Q Q Q a,b a,b Q T I Roos of Q ae Eigenvales of T

44 Eemlay Tansfomaions

45 Eemlay Tansfomaion ~ ~ ~ T w w

46 onsc T given Eigenvecos Wan T = T = asic nswe T = a b How i woks fo T = a b

47 Wan onsc T given wo diffeen o oins T = ~ T = ~ asic nswe T = a ~ b ~ Woks fo and T = a ~ T = b ~

48 Thid oin T = ~ T = a ~ b ~ T = a ~ b ~ Pick a and b o make ~ = a ~ b ~ ~ ~ ~ = + ~ ~ ~ ~ ~

49 The answe a = ~ ~ b= ~ ~ T = ~ ~ ~ + ~ ~ ~

50 How i woks T = ~ ~ ~ + ~ ~ ~ T = ~ ~ ~ + ~ ~ ~ T = ~ ~ ~ + ~ ~ ~ T = ~ ~ ~ + ~ ~ ~

51 How i woks T = ~ ~ ~ + ~ ~ ~ T = ~ ~ ~ + ~ ~ ~ T = ~ ~ ~ + ~ ~ ~ T = ~ ~ ~ + ~ ~ ~ ~ ~ = ~

52 Ne T = ~ ~ ~ + ~ ~ ~ T = ~ ~ ~ T = ~ ~ ~ T = ~ ~ ~

53 eeminan T = ~ ~ ~ + ~ ~ ~ T T = ~ ~ ~ ~ ~ ~ eeminan is negaive eacly when necessay fo ode evesal n invaian of he oins,, Geomeic meaning of sign

54 Highe imensions T ~ ~ b d ~ b ~ d

55 Highe imensions T ~ ~ s ~ ~ s s T = a ~ + b ~ s g ~ s Woks fo,,s Now find a,b,g o make i wok fo

56 Highe imensions T ~ ~ s ~ s s s s s s s s T = ~ ~ + ~ ~ ~ ~

57 o Poins in P 1 w

58 Ineleaving of o Poins Same ineleaving w iffeen ineleaving w

59 Thee ossible ineleavings w w w

60 iagams fo o Poins w w w w 1 V V 3 V

61 iagams fo o Poins V1 V V 3 0 V1 V V3 0

62 oss Raio bsole Invaian V V 3 old ick any of V V V V V V 1,, as coss aio Relaionshi beween V V 1 V V 1 V V V V V V V V

63 3 view of invaian sace V1 V V3 0 V1 V V 3 V 3 V 1 V V 1 Homogeneosly scale o nomalize ono ni cicle V V V V 3 V 1 cos V 1

64 3 view of invaian sace V1 V V3 0 V 1 V 3 V 1 V 3 V V V 3 V 1 V V 3 V 1 V

65 eemine which sign indicaes which ineleaving w w 1 V V 3 V w w

66 ollow in V sace V V V V V V V 1 V V V 1 V 1 V 1 V 3 V V 3 V V 3 V

67 ollow in V sace V 1 V V V V V 1 3 V 1 1 V 1 V V 1 V 0 V V 1 V 1 V V 1 1 V 0 V 1 3 V 1 V 1 V 1 V 1 V 1 V 1 V 3 V V 3 V V 3 V V 3 V V 3 V V 3 V V 3 V V1

68 ollow in V sace V 1 V V V V V 1 3 V 1 1 V 1 V V 1 V 0 V V 1 V 1 V V 1 1 V 0 V 1 3 an ansfom o an ansfom o V 1 V 1 V 1 V 1 V 1 V 1 V 3 V V 3 V V 3 V V 3 V V 3 V V 3 V Hamonic Se V V 3 V V1 V 3

69 eemine which sign indicaes which ineleaving w ^ ^ ^ ^ V w V ^ ^ V w V w 1 V V w 3

70 Signs and ineleaving Sign(V 1,V,V 3 ) Odeing of oins on P 1 View in (,w) lane

71 ee Ineleaving Tes Old way V w 1 V V w 3 ee way VV 1 w w V V w V V w w w

72 iagams V 1 V = V V 3 = = V 3 V 1 =

73 es Ineleaving Tes w 0 Û w 0 Û 0 Û

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