1 00:00:05,640 --> 00:00:09,369 In this video I'm going to explain what's meant by a 2 00:00:09,369 --> 00:00:12,420 symmetric matrix and the transpose of a matrix. Let's 3 00:00:12,420 --> 00:00:14,454 have a look at a matrix. 4 00:00:16,000 --> 00:00:20,800 This one I'm going to call M and it's the Matrix 4. 5 00:00:21,540 --> 00:00:25,120 Minus 1 -- 1 nine. 6 00:00:26,650 --> 00:00:29,686 What I'd like to do is focus on the leading diagonal. Remember, 7 00:00:29,686 --> 00:00:32,722 the leading diagonal is the one from top left to bottom right. 8 00:00:32,722 --> 00:00:33,734 That's this one here. 9 00:00:34,800 --> 00:00:38,518 And if we imagine that this is a mirror, you'll see that there's 10 00:00:38,518 --> 00:00:41,378 a mirror image across across this leading diagonal of these 11 00:00:41,378 --> 00:00:44,810 elements. The element minus one. Here is the same as that here, 12 00:00:44,810 --> 00:00:47,956 and a matrix with that property is called a symmetric matrix. 13 00:00:49,110 --> 00:00:51,234 Let's have a look at a slightly larger one. 14 00:00:53,970 --> 00:01:00,306 This one will be a three by three matrix. Let's suppose it's 15 00:01:00,306 --> 00:01:02,418 got the elements 273794. 16 00:01:03,280 --> 00:01:09,049 347 and again focus on the leading diagonal here. 17 00:01:10,230 --> 00:01:12,957 And look across the leading diagonal at the particular 18 00:01:12,957 --> 00:01:15,078 elements. We've got a 7 and A7. 19 00:01:15,690 --> 00:01:19,364 Three and three and four and four, so this leading diagonal 20 00:01:19,364 --> 00:01:23,372 acts a little bit like a mirror line. It's a line of 21 00:01:23,372 --> 00:01:27,380 symmetry, so both this matrix N and the previous one M are 22 00:01:27,380 --> 00:01:28,382 called symmetric matrices. 23 00:01:40,410 --> 00:01:43,914 One thing I'd like to do now is introduce what's called the 24 00:01:43,914 --> 00:01:47,418 transpose of the matrix. If we take any matrix A, for example. 25 00:01:48,980 --> 00:01:54,141 So let's go with the Matrix A from before 4 -- 113 nine. 26 00:01:56,280 --> 00:02:01,768 If we look at the first row 4 -- 1 and we form a new matrix 27 00:02:01,768 --> 00:02:06,227 where the first column is, this row 4 -- 1, so the first 28 00:02:06,227 --> 00:02:07,942 column is 4 -- 1. 29 00:02:10,250 --> 00:02:14,280 And this row here 13 nine becomes the second column 30 00:02:14,280 --> 00:02:18,713 here 13 nine we say that this new matrix here is 31 00:02:18,713 --> 00:02:21,937 obtained by taking the transpose of the original 32 00:02:21,937 --> 00:02:23,952 matrix, and we call this. 33 00:02:25,220 --> 00:02:29,191 The transpose matrix and we denote it by a with a 34 00:02:29,191 --> 00:02:32,801 superscript T for transpose, so this matrix A transpose is 35 00:02:32,801 --> 00:02:36,050 the transpose of this. It's obtained by interchanging the 36 00:02:36,050 --> 00:02:39,660 rows and columns. So the first row becomes the first 37 00:02:39,660 --> 00:02:42,548 column, the 2nd row becomes the second column. 38 00:02:44,810 --> 00:02:48,898 If we look at the matrix M We started with here and try and 39 00:02:48,898 --> 00:02:51,818 find the transpose of it. Let's do that up here. 40 00:02:54,270 --> 00:02:57,285 The transpose of this matrix is obtained by interchanging 41 00:02:57,285 --> 00:03:01,640 the rows and columns. So the first row 4 -- 1 becomes the 42 00:03:01,640 --> 00:03:02,310 first column. 43 00:03:03,740 --> 00:03:08,667 And the 2nd row minus 19 becomes the second column you'll see in 44 00:03:08,667 --> 00:03:12,836 this particular case that the matrix M and the matrix M 45 00:03:12,836 --> 00:03:17,005 transpose are the same. So if you have a symmetric matrix, 46 00:03:17,005 --> 00:03:21,932 it's the same as its transpose. The same will be true of matrix 47 00:03:21,932 --> 00:03:26,480 N here, which you can verify for yourself. So symmetric matrix is 48 00:03:26,480 --> 00:03:31,407 actually one which is which has the property that a is equal to 49 00:03:31,407 --> 00:03:32,923 its transpose. And that's 50 00:03:32,923 --> 00:03:36,174 another definition. Of what we mean by a symmetric matrix. 51 00:03:37,400 --> 00:03:41,470 Let's have a look at another example of finding the transpose 52 00:03:41,470 --> 00:03:45,910 of a matrix. We can find the transpose of any matrix. It 53 00:03:45,910 --> 00:03:50,720 doesn't have to be a square matrix. Let's have a look at an 54 00:03:50,720 --> 00:03:54,790 example such as finding the transpose of matrix C, which was 55 00:03:54,790 --> 00:03:57,380 seven 1 -- 3, two, 4, four. 56 00:03:59,820 --> 00:04:03,510 Note that this is a three row two column matrix. 57 00:04:05,160 --> 00:04:08,735 When we find it's transpose, what we do is we take the first 58 00:04:08,735 --> 00:04:12,870 row. And it becomes the first column in the transpose matrix. 59 00:04:14,980 --> 00:04:19,751 The 2nd row becomes the second column and the final Row 4, four 60 00:04:19,751 --> 00:04:23,421 becomes the final column. So what we've done is we've 61 00:04:23,421 --> 00:04:27,091 interchanged the rows and the columns to form the transpose, 62 00:04:27,091 --> 00:04:31,862 and this one we would denote as C with a superscript T for 63 00:04:31,862 --> 00:04:35,899 transpose. And note that in this case the resulting matrix now 64 00:04:35,899 --> 00:04:40,670 has got two rows and three columns, so this is a two by 65 00:04:40,670 --> 00:04:44,707 three matrix and this results is true in general as well. 66 00:04:44,990 --> 00:04:47,465 We find a transpose by interchanging the rows and 67 00:04:47,465 --> 00:04:51,040 columns. You'll find that, say, three by two becomes a 2 by 3. 68 00:04:52,110 --> 00:04:57,091 Four by three will become a 3 by 4 and M by N would become an N 69 00:04:57,091 --> 00:04:58,556 by M and so on.