SE590 Lece 4 Moe abo P 1 Tansfoming Tansfomaions James. linn Jimlinn.om h://coses.cs.washingon.ed/coses/cse590b/13a/
Peviosly On SE590b
Tansfomaions M M w w w w w
The ncion w w w w w w 0 w w 0 w 0 w
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
New oodinae Sysem E G H G H E E E G H G H E H G E 4 d 4 G H 1 4 1 4 E H G E d H G
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
Roadma of M oaions singla ideniy Niloen d = 0 Single eigenvale Involions (ace=0)
nd Now
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
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
Tansfomaion of Tansfomaions s s s v s v s v v v vv E G H G H E 1 1 0 0 E 1 1 0 0 0 0 1 1 G 0 0 1 1 H E v s 0 0 0 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
isill wih Roaion Tansfom E v s 0 0 0 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 0 1 1 H 0 E 1 0 0 0 E 0 cos sin 0 G 0 sin cos 0 G H 0 0 0 1 H Simila o wha we did wih Q: E E
Roaion Tansfom Roae o make G zeo E 1 0 0 0 E 0 cos sin 0 0 0 sin cos 0 G H 0 0 0 1 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
Roaion Tansfom Roae o make G zeo E 1 0 0 0 E 0 cos sin 0 0 0 sin cos 0 G H 0 0 0 1 H Pojec ono ni shee E H 1 4 d H H H 1 E H d E
Maings Qadaic Polys (3) /E oae E ojec /E E Tansfomaion m (4) H oae [E,,G,H] ojec E
Sign lis Roaing 90 degees flis (,G) sign E 1 0 0 0 E 0 1 0 1 0 0 1 0 G 0 0 1 0 G H 0 0 0 1 H Scaling by -1 in flis (E,) signs E 1 0 0 0 E 1 0 0 1 0 0 0 1 G 0 0 1 0 G H 0 0 0 1 H Echanging,w flis (E,G) signs d E H E 1 0 0 0 E 0 1 0 1 0 0 1 0 G 0 0 1 0 G H 0 0 0 1 H
Sign lis H Echanging,w flis (E,G) signs E 1 0 0 0 E 0 1 0 1 0 0 1 0 G 0 0 1 0 G H 0 0 0 1 H d E
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 / 10 170 10
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
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
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
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
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
isilled EGH Sace H 1 0 0 1 0 0 E E 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 0 0 1 0 1 1 0 0 1 1 0 E H H E 1 0 0 0
Phase gm and isilled M M 0 M M M M 0 E 0 1 0 0 1 0 E Niloen singla d 1 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 1 1 1 0 1 1 0 H M M M M M M E H H E d M M M M
Phase gm and isilled Q M M 0 M M M M 0 E 0 1 0 0 1 0 E I E Niloen singla M M M M M M d 1 1 1 1 0 1 1 0 1 1 0 1 0 0 1 isingish beween 1 1 0 1 1 1 1 V R H 1 1 0 0 1 1 0 d M M M M M V R H V R M V R M M V R
Phase gm and isilled M M 0 M M M M 0 E 0 1 0 0 1 0 E Niloen singla d 1 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 1 1 1 0 1 1 0 H M M M M M M d M M M M M V R H Use if d 0
Inenal Sce
Using oe odcs o make M M = k n 1 0 0 1 1 0 0 1
Wie k,n in ems of, M = k n k = a + b n = c + d M = a b c d
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 ~ ~
Pick nice, = 0 1 = 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 + M ~ ~ ~ + ~ ~ ~ ~ ~ ~
Niloen E 0 1 0 0 1 0 E 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 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
Niloen 0 1 0 0 N N N w w a N a N H d
Idemoen 0 0 0 1 H a a 0 w w d
n Ideniy H/E ^ I G/E /E
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
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
Roaion w E H H E E w + H E ^I + H H d
Roaion Involion E H, E 0 H E + H w w H d
Single eigenvale d = 0 w 1 1 0 1 w ^I H d
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
Eemlay Tansfomaions
Eemlay Tansfomaion ~ ~ ~ T w w
onsc T given Eigenvecos Wan T = T = asic nswe T = a b How i woks fo T = a b
Wan onsc T given wo diffeen o oins T = ~ T = ~ asic nswe T = a ~ b ~ Woks fo and T = a ~ T = b ~
Thid oin T = ~ T = a ~ b ~ T = a ~ b ~ Pick a and b o make ~ = a ~ b ~ ~ ~ ~ = + ~ ~ ~ ~ ~
The answe a = ~ ~ b= ~ ~ T = ~ ~ ~ + ~ ~ ~
How i woks T = ~ ~ ~ + ~ ~ ~ T = ~ ~ ~ + ~ ~ ~ T = ~ ~ ~ + ~ ~ ~ T = ~ ~ ~ + ~ ~ ~
How i woks T = ~ ~ ~ + ~ ~ ~ T = ~ ~ ~ + ~ ~ ~ T = ~ ~ ~ + ~ ~ ~ T = ~ ~ ~ + ~ ~ ~ ~ ~ = ~
Ne T = ~ ~ ~ + ~ ~ ~ T = ~ ~ ~ T = ~ ~ ~ T = ~ ~ ~
eeminan T = ~ ~ ~ + ~ ~ ~ T T = ~ ~ ~ ~ ~ ~ eeminan is negaive eacly when necessay fo ode evesal n invaian of he oins,, Geomeic meaning of sign
Highe imensions T ~ ~ b d ~ b ~ d
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
Highe imensions T ~ ~ s ~ s s s s s s s s T = ~ ~ + ~ ~ ~ ~
o Poins in P 1 w
Ineleaving of o Poins Same ineleaving w iffeen ineleaving w
Thee ossible ineleavings w w w
iagams fo o Poins w w w w 1 V V 3 V
iagams fo o Poins V1 V V 3 0 V1 V V3 0
oss Raio bsole Invaian V V 3 old ick any of V V V V V V 1,, 3 3 1 as coss aio Relaionshi beween V V 1 V V 1 V V V 1 3 1 V V V V V 3 1 1 1 1 1
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 1 3 1 V V V V 3 V 1 cos V 1
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
eemine which sign indicaes which ineleaving 1 1 0 1 1 1 w w 1 V V 3 V w w
ollow in V sace V V V 1 3 1 0 1 1 V V V 1 3 0 1 1 1 3 V 1 V V 1 3 1 V 1 V 1 V 1 V 3 V V 3 V V 3 V
ollow in V sace 1 3 1 0 V 1 V V 1 1 3 V 0 1 1 V V 1 3 V 1 1 V 1 V 3 1 3 0 1 V 1 V 0 V 1 1 3 V 1 V 1 V 1 3 1 1 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
ollow in V sace 1 3 1 0 V 1 V V 1 1 3 V 0 1 1 V V 1 3 V 1 1 V 1 V 3 1 3 0 1 V 1 V 0 V 1 1 3 V 1 V 1 V 1 3 1 1 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
eemine which sign indicaes which ineleaving 0 1 1 1 w 1 1 1 3 ^ ^ ^ ^ V w V ^ ^ V w V w 1 V V w 3
Signs and ineleaving Sign(V 1,V,V 3 ) Odeing of oins on P 1 View in (,w) lane
ee Ineleaving Tes Old way V w 1 V V w 3 ee way VV 1 w w V V w 3 3 1 V V w w w
iagams V 1 V = V V 3 = = V 3 V 1 =
es Ineleaving Tes w 0 Û w 0 Û 0 Û