0:00 Introduction

1:13 Example

2:00 Row exchange

2:30 Gaussian elimination

4:20 Next row exchange

5:45 Last step

1 00:00:00,400 –> 00:00:02,150 Hello and welcome to this

2 00:00:02,160 –> 00:00:03,819 video about linear algebra.

3 00:00:03,950 –> 00:00:05,230 And as always many, many

4 00:00:05,239 –> 00:00:06,800 thanks to all the nice people

5 00:00:06,809 –> 00:00:07,989 that support me on Steady

6 00:00:08,000 –> 00:00:08,819 or paypal.

7 00:00:09,210 –> 00:00:10,699 If you want a PDF version

8 00:00:10,710 –> 00:00:12,329 of this video, the link is

9 00:00:12,340 –> 00:00:13,779 in the description. Today’s

10 00:00:13,789 –> 00:00:15,239 topic is the so-called

11 00:00:15,250 –> 00:00:16,639 plu decomposition

12 00:00:16,649 –> 00:00:17,299 for matrices.

13 00:00:17,969 –> 00:00:19,780 You can see this as a supplemental

14 00:00:19,790 –> 00:00:21,270 video if you have already

15 00:00:21,280 –> 00:00:22,780 watched my video about the

16 00:00:22,790 –> 00:00:23,979 LU decomposition.

17 00:00:24,719 –> 00:00:26,209 In other words, this is just

18 00:00:26,219 –> 00:00:27,280 a generalization.

19 00:00:27,719 –> 00:00:29,600 However, L still stands for

20 00:00:29,610 –> 00:00:31,090 a lower triangular matrix.

21 00:00:31,100 –> 00:00:33,000 Hence it’s always a square

22 00:00:33,009 –> 00:00:34,709 matrix but

23 00:00:34,720 –> 00:00:36,349 U in general is not a

24 00:00:36,360 –> 00:00:38,040 square matrix because it

25 00:00:38,049 –> 00:00:39,360 always needs to be of the

26 00:00:39,369 –> 00:00:41,159 same size as the original

27 00:00:41,169 –> 00:00:42,959 matrix we want to decompose.

28 00:00:43,419 –> 00:00:44,759 So what we want to get out

29 00:00:44,770 –> 00:00:46,290 here is this so-called row

30 00:00:46,380 –> 00:00:47,529 echelon form.

31 00:00:47,810 –> 00:00:49,259 Please note for a square

32 00:00:49,270 –> 00:00:50,750 matrix, this would be an

33 00:00:50,759 –> 00:00:52,240 upper triangular matrix.

34 00:00:52,250 –> 00:00:53,720 So we still denote it by

35 00:00:53,729 –> 00:00:54,099 U.

36 00:00:54,770 –> 00:00:56,080 And the last ingredient we

37 00:00:56,090 –> 00:00:57,709 have here is the matrix P

38 00:00:57,720 –> 00:00:59,290 which saves all the row

39 00:00:59,299 –> 00:01:00,729 exchanges we have to do.

40 00:01:01,150 –> 00:01:02,389 Therefore, this is what we

41 00:01:02,400 –> 00:01:04,309 call a permutation matrix.

42 00:01:04,510 –> 00:01:04,870 OK.

43 00:01:04,879 –> 00:01:06,209 Now you know the three parts

44 00:01:06,220 –> 00:01:07,269 of the decomposition.

45 00:01:07,279 –> 00:01:08,949 So let’s calculate them for

46 00:01:08,959 –> 00:01:10,709 an example, I want the

47 00:01:10,720 –> 00:01:12,610 matrix A to be a four times

48 00:01:12,620 –> 00:01:13,669 five matrix

49 00:01:14,440 –> 00:01:15,440 there we have it.

50 00:01:15,449 –> 00:01:16,730 This is the matrix I have

51 00:01:16,739 –> 00:01:17,760 chosen for today.

52 00:01:18,220 –> 00:01:19,580 Now, if you want to start

53 00:01:19,589 –> 00:01:21,120 here with your common LU

54 00:01:21,129 –> 00:01:22,739 decomposition, you immediately

55 00:01:22,750 –> 00:01:23,739 find the problem.

56 00:01:24,250 –> 00:01:25,809 You can’t use the zero as

57 00:01:25,819 –> 00:01:27,599 a pivot to eliminate all

58 00:01:27,610 –> 00:01:28,639 the other numbers in the

59 00:01:28,650 –> 00:01:29,500 first column.

60 00:01:29,790 –> 00:01:31,000 Hence, the first step you

61 00:01:31,010 –> 00:01:32,569 need to do is to go through

62 00:01:32,580 –> 00:01:34,550 the column to find a nonzero

63 00:01:34,559 –> 00:01:36,279 element to use as a pivot.

64 00:01:37,309 –> 00:01:37,720 OK.

65 00:01:37,730 –> 00:01:39,279 We find it in the second row.

66 00:01:39,290 –> 00:01:40,910 Therefore, we need to exchange

67 00:01:40,919 –> 00:01:42,180 the first and the second

68 00:01:42,190 –> 00:01:42,550 row.

69 00:01:42,980 –> 00:01:44,620 Now a permutation matrix

70 00:01:44,629 –> 00:01:46,489 that exchanges row one with

71 00:01:46,500 –> 00:01:48,339 row two looks like this.

72 00:01:48,989 –> 00:01:50,220 It’s simply a four times

73 00:01:50,230 –> 00:01:51,769 four identity matrix where

74 00:01:51,779 –> 00:01:53,160 we flip the two rows, we’re

75 00:01:53,169 –> 00:01:55,089 interested in an important

76 00:01:55,099 –> 00:01:56,940 property is now if we square

77 00:01:56,949 –> 00:01:58,839 that matrix, we get out our

78 00:01:58,849 –> 00:02:00,220 identity matrix.

79 00:02:00,349 –> 00:02:01,750 Hence, this is our first

80 00:02:01,760 –> 00:02:02,120 step.

81 00:02:02,129 –> 00:02:03,480 We just apply the identity

82 00:02:03,489 –> 00:02:04,629 matrix on the left.

83 00:02:05,199 –> 00:02:06,580 Then in the next step, we

84 00:02:06,589 –> 00:02:08,130 multiply one of the two

85 00:02:08,139 –> 00:02:09,839 matrices to the right, which

86 00:02:09,850 –> 00:02:11,289 means we exchange the two

87 00:02:11,300 –> 00:02:11,940 rows here.

88 00:02:12,639 –> 00:02:14,449 Afterwards, we have our pivot

89 00:02:14,460 –> 00:02:15,649 at the correct position.

90 00:02:16,330 –> 00:02:17,800 This means that we now can

91 00:02:17,809 –> 00:02:19,539 put the L matrix into the

92 00:02:19,550 –> 00:02:19,940 game.

93 00:02:20,360 –> 00:02:21,470 It should be a four times

94 00:02:21,479 –> 00:02:22,300 four matrix.

95 00:02:22,309 –> 00:02:24,130 So we start as usual with

96 00:02:24,139 –> 00:02:25,220 the identity matrix.

97 00:02:25,990 –> 00:02:27,110 And now we have to do the

98 00:02:27,119 –> 00:02:28,649 Gaussian elimination in the

99 00:02:28,660 –> 00:02:29,529 first column.

100 00:02:30,089 –> 00:02:31,710 So we want to generate zeros

101 00:02:31,720 –> 00:02:33,419 below the pivot which means

102 00:02:33,429 –> 00:02:34,970 that we are already finished

103 00:02:34,979 –> 00:02:35,410 here.

104 00:02:35,949 –> 00:02:37,699 In other words, second row

105 00:02:37,710 –> 00:02:39,559 minus zero (times) the first

106 00:02:39,570 –> 00:02:41,490 row. And the zero

107 00:02:41,500 –> 00:02:43,410 then goes into the L matrix.

108 00:02:44,100 –> 00:02:45,210 So this was easy.

109 00:02:45,220 –> 00:02:46,910 So let’s go to the next number

110 00:02:46,919 –> 00:02:47,919 which is two.

111 00:02:48,320 –> 00:02:50,160 So third row minus

112 00:02:50,169 –> 00:02:52,000 two times the first row

113 00:02:52,529 –> 00:02:54,270 and the multiple we subtract

114 00:02:54,279 –> 00:02:56,139 is the number that goes into

115 00:02:56,149 –> 00:02:57,130 the L matrix.

116 00:02:57,600 –> 00:02:58,850 Also not so hard.

117 00:02:58,860 –> 00:03:00,289 The only number that changes

118 00:03:00,300 –> 00:03:01,649 is the five that gets to

119 00:03:01,660 –> 00:03:02,210 a three.

120 00:03:02,639 –> 00:03:03,850 And the last number in the

121 00:03:03,860 –> 00:03:05,149 column is the one.

122 00:03:05,479 –> 00:03:06,949 So we subtract one.

123 00:03:07,369 –> 00:03:08,639 And as before, this is the

124 00:03:08,649 –> 00:03:10,110 number that goes into the

125 00:03:10,119 –> 00:03:10,910 L matrix.

126 00:03:11,470 –> 00:03:13,380 And with that, we are finished

127 00:03:13,389 –> 00:03:15,339 with the first Gaussian elimination.

128 00:03:16,029 –> 00:03:17,100 Then in the next step, we

129 00:03:17,110 –> 00:03:18,490 go to the next column and

130 00:03:18,500 –> 00:03:19,729 choose the next pivot.

131 00:03:19,740 –> 00:03:21,449 And we see it’s nonzero.

132 00:03:21,460 –> 00:03:22,860 So we don’t need any row

133 00:03:22,869 –> 00:03:23,869 exchanges here.

134 00:03:24,449 –> 00:03:25,660 Otherwise we do the same

135 00:03:25,669 –> 00:03:27,110 as before we generate

136 00:03:27,119 –> 00:03:28,679 zeros below the pivot.

137 00:03:29,380 –> 00:03:30,550 In the first step, we just

138 00:03:30,559 –> 00:03:31,809 have to subtract one.

139 00:03:32,500 –> 00:03:34,300 And as before this one goes

140 00:03:34,309 –> 00:03:36,279 immediately to our L matrix.

141 00:03:36,910 –> 00:03:38,000 On the other side, we get

142 00:03:38,009 –> 00:03:39,690 a lot of zeros and one

143 00:03:39,699 –> 00:03:40,039 here.

144 00:03:40,770 –> 00:03:41,869 And then in the next step,

145 00:03:41,880 –> 00:03:43,410 we have to subtract two times

146 00:03:43,419 –> 00:03:44,240 the second row.

147 00:03:44,710 –> 00:03:46,070 This too then goes as

148 00:03:46,080 –> 00:03:48,050 always into the L matrix.

149 00:03:48,550 –> 00:03:49,789 And with that calculation,

150 00:03:49,800 –> 00:03:50,869 we are finished with the

151 00:03:50,880 –> 00:03:52,500 second column, let’s go to

152 00:03:52,509 –> 00:03:53,270 the third column.

153 00:03:53,929 –> 00:03:54,389 OK.

154 00:03:54,399 –> 00:03:56,100 So this is not a pivot,

155 00:03:56,279 –> 00:03:57,949 but this is also not a pivot.

156 00:03:58,000 –> 00:03:59,639 So there are no pivots in

157 00:03:59,649 –> 00:04:00,550 the third column.

158 00:04:01,229 –> 00:04:02,690 This means that this column

159 00:04:02,699 –> 00:04:04,279 is finished, we have to go

160 00:04:04,289 –> 00:04:05,220 to the next column.

161 00:04:05,929 –> 00:04:07,580 However, there we also find

162 00:04:07,589 –> 00:04:09,460 a zero which is not a pivot,

163 00:04:09,570 –> 00:04:11,199 but there is another pivot

164 00:04:11,210 –> 00:04:11,800 below.

165 00:04:12,889 –> 00:04:14,029 And with that, we know what

166 00:04:14,039 –> 00:04:15,470 to do, we have to do a row

167 00:04:15,479 –> 00:04:16,190 exchange.

168 00:04:16,980 –> 00:04:18,358 In order to do that, we will

169 00:04:18,369 –> 00:04:19,890 insert the identity matrix

170 00:04:19,899 –> 00:04:20,298 again.

171 00:04:21,029 –> 00:04:22,869 So this is again a permutation

172 00:04:22,880 –> 00:04:24,559 matrix squared where we have

173 00:04:24,570 –> 00:04:26,200 now the permutation matrix,

174 00:04:26,209 –> 00:04:27,920 third row to

175 00:04:27,929 –> 00:04:29,589 fourth row, of

176 00:04:29,600 –> 00:04:30,880 course, one of them, we can

177 00:04:30,890 –> 00:04:32,119 apply to the right hand side

178 00:04:32,130 –> 00:04:34,040 again to get our row exchange.

179 00:04:34,549 –> 00:04:35,760 Indeed, there we get what

180 00:04:35,769 –> 00:04:36,250 we want.

181 00:04:36,260 –> 00:04:37,720 Now we have to pivot at the

182 00:04:37,730 –> 00:04:38,600 correct position.

183 00:04:39,350 –> 00:04:40,839 However, the permutation

184 00:04:40,850 –> 00:04:42,230 matrix is not at the correct

185 00:04:42,239 –> 00:04:43,130 position yet.

186 00:04:43,160 –> 00:04:44,399 We need that on the left

187 00:04:44,440 –> 00:04:45,019 hand side.

188 00:04:45,820 –> 00:04:47,390 Therefore, in the next step,

189 00:04:47,399 –> 00:04:48,859 we will also add the identity

190 00:04:48,869 –> 00:04:50,589 matrix on the left hand side.

191 00:04:51,299 –> 00:04:52,679 So here again, we have the

192 00:04:52,690 –> 00:04:54,320 permutation matrix squared.

193 00:04:55,119 –> 00:04:56,709 Now, you know, when we apply

194 00:04:56,720 –> 00:04:58,369 this matrix to the right,

195 00:04:58,410 –> 00:04:59,929 we will exchange these two

196 00:04:59,940 –> 00:05:00,660 rows here.

197 00:05:01,089 –> 00:05:02,679 However, if we apply this

198 00:05:02,690 –> 00:05:04,480 matrix to the left, we will

199 00:05:04,489 –> 00:05:05,799 exchange the corresponding

200 00:05:05,809 –> 00:05:06,459 columns.

201 00:05:07,220 –> 00:05:09,140 Nevertheless, this is exactly

202 00:05:09,149 –> 00:05:10,790 what we do in the next step.

203 00:05:10,799 –> 00:05:12,470 First exchanging the

204 00:05:12,480 –> 00:05:14,029 rows looks like this.

205 00:05:14,690 –> 00:05:16,589 And afterwards, we exchanged

206 00:05:16,600 –> 00:05:18,450 the columns which

207 00:05:18,459 –> 00:05:19,850 results in this

208 00:05:19,859 –> 00:05:20,609 matrix.

209 00:05:21,140 –> 00:05:22,690 The good thing is you see

210 00:05:22,700 –> 00:05:24,170 we get again a lower

211 00:05:24,179 –> 00:05:25,399 triangular matrix

212 00:05:26,040 –> 00:05:27,709 but don’t oversee that this

213 00:05:27,720 –> 00:05:29,160 really changes the matrix

214 00:05:29,170 –> 00:05:30,170 L from before.

215 00:05:30,339 –> 00:05:31,529 The good thing is you can

216 00:05:31,540 –> 00:05:33,059 remember the whole operation

217 00:05:33,070 –> 00:05:34,269 by just doing the row

218 00:05:34,390 –> 00:05:36,070 exchange in the blue part

219 00:05:36,079 –> 00:05:36,420 here.

220 00:05:37,059 –> 00:05:38,329 However, of course, now you

221 00:05:38,339 –> 00:05:39,500 have seen where it really

222 00:05:39,510 –> 00:05:40,179 comes from.

223 00:05:40,959 –> 00:05:41,299 OK.

224 00:05:41,309 –> 00:05:42,779 Now that everything is in

225 00:05:42,790 –> 00:05:44,670 the right order, we can continue

226 00:05:44,679 –> 00:05:45,540 our procedure.

227 00:05:46,250 –> 00:05:47,510 This means that we want to

228 00:05:47,519 –> 00:05:49,269 generate a zero here, but

229 00:05:49,279 –> 00:05:50,510 it’s already there, which

230 00:05:50,519 –> 00:05:51,790 means that we are finished

231 00:05:51,799 –> 00:05:53,640 with this column in the

232 00:05:53,649 –> 00:05:55,000 next column, we then find

233 00:05:55,010 –> 00:05:56,660 the pivot which is here,

234 00:05:56,739 –> 00:05:58,079 but we don’t have to generate

235 00:05:58,089 –> 00:05:59,070 zeros anymore.

236 00:05:59,630 –> 00:06:01,320 Indeed, this was our last

237 00:06:01,329 –> 00:06:02,859 step in the whole calculation.

238 00:06:02,890 –> 00:06:04,559 We have found the row echelon

239 00:06:04,570 –> 00:06:05,760 form here on the right hand

240 00:06:05,769 –> 00:06:06,160 side.

241 00:06:06,910 –> 00:06:07,320 OK.

242 00:06:07,329 –> 00:06:08,739 So there we have our matrix

243 00:06:08,869 –> 00:06:09,410 U.

244 00:06:10,339 –> 00:06:12,119 And this is our matrix

245 00:06:12,130 –> 00:06:14,109 L and the

246 00:06:14,119 –> 00:06:15,279 last part is what we still

247 00:06:15,290 –> 00:06:16,299 can multiply.

248 00:06:16,309 –> 00:06:17,970 But then we get our matrix

249 00:06:17,980 –> 00:06:18,429 P.

250 00:06:19,339 –> 00:06:20,489 We’ve reached our goal.

251 00:06:20,500 –> 00:06:22,450 Our matrix A from the beginning

252 00:06:22,459 –> 00:06:24,320 is given as P times

253 00:06:24,329 –> 00:06:26,290 L times U and

254 00:06:26,299 –> 00:06:27,750 therefore we call this the

255 00:06:27,760 –> 00:06:29,279 PLU decomposition.

256 00:06:30,059 –> 00:06:30,429 OK?

257 00:06:30,440 –> 00:06:31,489 I think that’s good enough

258 00:06:31,500 –> 00:06:32,720 for this example.

259 00:06:32,779 –> 00:06:34,279 I hope you can now apply

260 00:06:34,290 –> 00:06:35,880 the whole procedure to another

261 00:06:35,890 –> 00:06:37,790 matrix A and with

262 00:06:37,799 –> 00:06:39,329 that, thanks for listening

263 00:06:39,339 –> 00:06:40,470 and see you next time.

264 00:06:40,480 –> 00:06:41,089 Bye.