WEBVTT 00:00:05.640 --> 00:00:09.369 In this video I'm going to explain what's meant by a 00:00:09.369 --> 00:00:12.420 symmetric matrix and the transpose of a matrix. Let's 00:00:12.420 --> 00:00:14.454 have a look at a matrix. 00:00:16.000 --> 00:00:20.800 This one I'm going to call M and it's the Matrix 4. 00:00:21.540 --> 00:00:25.120 Minus 1 -- 1 nine. 00:00:26.650 --> 00:00:29.686 What I'd like to do is focus on the leading diagonal. Remember, 00:00:29.686 --> 00:00:32.722 the leading diagonal is the one from top left to bottom right. 00:00:32.722 --> 00:00:33.734 That's this one here. 00:00:34.800 --> 00:00:38.518 And if we imagine that this is a mirror, you'll see that there's 00:00:38.518 --> 00:00:41.378 a mirror image across across this leading diagonal of these 00:00:41.378 --> 00:00:44.810 elements. The element minus one. Here is the same as that here, 00:00:44.810 --> 00:00:47.956 and a matrix with that property is called a symmetric matrix. 00:00:49.110 --> 00:00:51.234 Let's have a look at a slightly larger one. 00:00:53.970 --> 00:01:00.306 This one will be a three by three matrix. Let's suppose it's 00:01:00.306 --> 00:01:02.418 got the elements 273794. 00:01:03.280 --> 00:01:09.049 347 and again focus on the leading diagonal here. 00:01:10.230 --> 00:01:12.957 And look across the leading diagonal at the particular 00:01:12.957 --> 00:01:15.078 elements. We've got a 7 and A7. 00:01:15.690 --> 00:01:19.364 Three and three and four and four, so this leading diagonal 00:01:19.364 --> 00:01:23.372 acts a little bit like a mirror line. It's a line of 00:01:23.372 --> 00:01:27.380 symmetry, so both this matrix N and the previous one M are 00:01:27.380 --> 00:01:28.382 called symmetric matrices. 00:01:40.410 --> 00:01:43.914 One thing I'd like to do now is introduce what's called the 00:01:43.914 --> 00:01:47.418 transpose of the matrix. If we take any matrix A, for example. 00:01:48.980 --> 00:01:54.141 So let's go with the Matrix A from before 4 -- 113 nine. 00:01:56.280 --> 00:02:01.768 If we look at the first row 4 -- 1 and we form a new matrix 00:02:01.768 --> 00:02:06.227 where the first column is, this row 4 -- 1, so the first 00:02:06.227 --> 00:02:07.942 column is 4 -- 1. 00:02:10.250 --> 00:02:14.280 And this row here 13 nine becomes the second column 00:02:14.280 --> 00:02:18.713 here 13 nine we say that this new matrix here is 00:02:18.713 --> 00:02:21.937 obtained by taking the transpose of the original 00:02:21.937 --> 00:02:23.952 matrix, and we call this. 00:02:25.220 --> 00:02:29.191 The transpose matrix and we denote it by a with a 00:02:29.191 --> 00:02:32.801 superscript T for transpose, so this matrix A transpose is 00:02:32.801 --> 00:02:36.050 the transpose of this. It's obtained by interchanging the 00:02:36.050 --> 00:02:39.660 rows and columns. So the first row becomes the first 00:02:39.660 --> 00:02:42.548 column, the 2nd row becomes the second column. 00:02:44.810 --> 00:02:48.898 If we look at the matrix M We started with here and try and 00:02:48.898 --> 00:02:51.818 find the transpose of it. Let's do that up here. 00:02:54.270 --> 00:02:57.285 The transpose of this matrix is obtained by interchanging 00:02:57.285 --> 00:03:01.640 the rows and columns. So the first row 4 -- 1 becomes the 00:03:01.640 --> 00:03:02.310 first column. 00:03:03.740 --> 00:03:08.667 And the 2nd row minus 19 becomes the second column you'll see in 00:03:08.667 --> 00:03:12.836 this particular case that the matrix M and the matrix M 00:03:12.836 --> 00:03:17.005 transpose are the same. So if you have a symmetric matrix, 00:03:17.005 --> 00:03:21.932 it's the same as its transpose. The same will be true of matrix 00:03:21.932 --> 00:03:26.480 N here, which you can verify for yourself. So symmetric matrix is 00:03:26.480 --> 00:03:31.407 actually one which is which has the property that a is equal to 00:03:31.407 --> 00:03:32.923 its transpose. And that's 00:03:32.923 --> 00:03:36.174 another definition. Of what we mean by a symmetric matrix. 00:03:37.400 --> 00:03:41.470 Let's have a look at another example of finding the transpose 00:03:41.470 --> 00:03:45.910 of a matrix. We can find the transpose of any matrix. It 00:03:45.910 --> 00:03:50.720 doesn't have to be a square matrix. Let's have a look at an 00:03:50.720 --> 00:03:54.790 example such as finding the transpose of matrix C, which was 00:03:54.790 --> 00:03:57.380 seven 1 -- 3, two, 4, four. 00:03:59.820 --> 00:04:03.510 Note that this is a three row two column matrix. 00:04:05.160 --> 00:04:08.735 When we find it's transpose, what we do is we take the first 00:04:08.735 --> 00:04:12.870 row. And it becomes the first column in the transpose matrix. 00:04:14.980 --> 00:04:19.751 The 2nd row becomes the second column and the final Row 4, four 00:04:19.751 --> 00:04:23.421 becomes the final column. So what we've done is we've 00:04:23.421 --> 00:04:27.091 interchanged the rows and the columns to form the transpose, 00:04:27.091 --> 00:04:31.862 and this one we would denote as C with a superscript T for 00:04:31.862 --> 00:04:35.899 transpose. And note that in this case the resulting matrix now 00:04:35.899 --> 00:04:40.670 has got two rows and three columns, so this is a two by 00:04:40.670 --> 00:04:44.707 three matrix and this results is true in general as well. 00:04:44.990 --> 00:04:47.465 We find a transpose by interchanging the rows and 00:04:47.465 --> 00:04:51.040 columns. You'll find that, say, three by two becomes a 2 by 3. 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 00:04:57.091 --> 00:04:58.556 by M and so on.