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Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:05.05,0:00:09.19,Default,,0000,0000,0000,,In this video I'm going to\Nexplain what is meant by a
Dialogue: 0,0:00:09.19,0:00:12.64,Default,,0000,0000,0000,,matrix and introduce the\Nnotation that we use when we're
Dialogue: 0,0:00:12.64,0:00:16.78,Default,,0000,0000,0000,,working with matrices. So let's\Nstart by looking at what we mean
Dialogue: 0,0:00:16.78,0:00:20.58,Default,,0000,0000,0000,,by a matrix and matrix is a\Nrectangular pattern of numbers.
Dialogue: 0,0:00:20.58,0:00:21.96,Default,,0000,0000,0000,,Let's have an example.
Dialogue: 0,0:00:23.44,0:00:25.95,Default,,0000,0000,0000,,I'm writing down a\Npattern of numbers.
Dialogue: 0,0:00:28.79,0:00:33.56,Default,,0000,0000,0000,,4 -- 113 and 9 and you see they\Nform a rectangular pattern and
Dialogue: 0,0:00:33.56,0:00:37.66,Default,,0000,0000,0000,,when we write down a matrix, we\Nusually enclose the numbers with
Dialogue: 0,0:00:37.66,0:00:41.75,Default,,0000,0000,0000,,some round brackets like that.\NSo that's our first example of a
Dialogue: 0,0:00:41.75,0:00:45.50,Default,,0000,0000,0000,,matrix. Let's have a look at\Nanother example which has a
Dialogue: 0,0:00:45.50,0:00:48.57,Default,,0000,0000,0000,,different size. So suppose we\Nhave the numbers 12.
Dialogue: 0,0:00:50.08,0:00:54.10,Default,,0000,0000,0000,,304 and again, this is a\Nrectangular pattern of
Dialogue: 0,0:00:54.10,0:00:58.57,Default,,0000,0000,0000,,numbers. I'll put them in\Nround brackets like that, and
Dialogue: 0,0:00:58.57,0:01:03.04,Default,,0000,0000,0000,,that's another example of a\Nmatrix. Let's have some more.
Dialogue: 0,0:01:05.56,0:01:07.25,Default,,0000,0000,0000,,71
Dialogue: 0,0:01:08.56,0:01:12.21,Default,,0000,0000,0000,,minus three to four and four.
Dialogue: 0,0:01:15.23,0:01:16.59,Default,,0000,0000,0000,,And a final example.
Dialogue: 0,0:01:18.14,0:01:20.75,Default,,0000,0000,0000,,A half 00.
Dialogue: 0,0:01:21.84,0:01:28.55,Default,,0000,0000,0000,,03000 and let's\Nsay nought .7.
Dialogue: 0,0:01:31.48,0:01:35.22,Default,,0000,0000,0000,,So here we have 4 examples\Nof matrices.
Dialogue: 0,0:01:37.38,0:01:41.41,Default,,0000,0000,0000,,They all have different sizes,\Nso let's look a little bit more
Dialogue: 0,0:01:41.41,0:01:45.78,Default,,0000,0000,0000,,about how we refer to the size\Nof a matrix. This first matrix
Dialogue: 0,0:01:45.78,0:01:50.48,Default,,0000,0000,0000,,here has got two rows and two\Ncolumns and we describe it as a
Dialogue: 0,0:01:50.48,0:01:55.19,Default,,0000,0000,0000,,two by two matrix. We write it\Nas two by two like that. That's
Dialogue: 0,0:01:55.19,0:01:59.56,Default,,0000,0000,0000,,just the size. When we write\Ndown the size of a matrix, we
Dialogue: 0,0:01:59.56,0:02:03.25,Default,,0000,0000,0000,,always give the number of rows\Nfirst and the columns second.
Dialogue: 0,0:02:03.25,0:02:05.94,Default,,0000,0000,0000,,So this has two rows and two\Ncolumns.
Dialogue: 0,0:02:07.55,0:02:08.94,Default,,0000,0000,0000,,What about this matrix?
Dialogue: 0,0:02:09.70,0:02:11.21,Default,,0000,0000,0000,,This is got one row.
Dialogue: 0,0:02:12.22,0:02:14.96,Default,,0000,0000,0000,,And 1234 columns.
Dialogue: 0,0:02:16.85,0:02:20.82,Default,,0000,0000,0000,,So this is a 1 by 4 matrix, one
Dialogue: 0,0:02:20.82,0:02:23.19,Default,,0000,0000,0000,,row. And four columns.
Dialogue: 0,0:02:25.86,0:02:29.21,Default,,0000,0000,0000,,This matrix has got 123 rows.
Dialogue: 0,0:02:31.61,0:02:32.91,Default,,0000,0000,0000,,And two columns.
Dialogue: 0,0:02:35.04,0:02:38.46,Default,,0000,0000,0000,,So it's a three by two\Nmatrix and the final example
Dialogue: 0,0:02:38.46,0:02:39.70,Default,,0000,0000,0000,,has got three rows.
Dialogue: 0,0:02:42.61,0:02:46.48,Default,,0000,0000,0000,,Three columns, so this is a\Nthree by three matrix, so
Dialogue: 0,0:02:46.48,0:02:50.35,Default,,0000,0000,0000,,remember that we always give\Nthe number of rows first and
Dialogue: 0,0:02:50.35,0:02:51.06,Default,,0000,0000,0000,,column 2nd.
Dialogue: 0,0:02:52.48,0:02:56.16,Default,,0000,0000,0000,,The other bit of notation that\Nwe'll need is that we often use
Dialogue: 0,0:02:56.16,0:02:59.56,Default,,0000,0000,0000,,a capital letter to denote a\Nmatrix, so we might call this
Dialogue: 0,0:02:59.56,0:03:00.69,Default,,0000,0000,0000,,first matrix here A.
Dialogue: 0,0:03:02.10,0:03:04.19,Default,,0000,0000,0000,,We might call the second 1B.
Dialogue: 0,0:03:05.38,0:03:06.51,Default,,0000,0000,0000,,This one C.
Dialogue: 0,0:03:08.02,0:03:09.14,Default,,0000,0000,0000,,And this 1D.
Dialogue: 0,0:03:10.87,0:03:14.83,Default,,0000,0000,0000,,So there we are four examples\Nof matrices, all of different
Dialogue: 0,0:03:14.83,0:03:19.51,Default,,0000,0000,0000,,sizes and we now know how to\Ndescribe a matrix in terms of
Dialogue: 0,0:03:19.51,0:03:23.47,Default,,0000,0000,0000,,the number of rows and the\Nnumber of columns that it
Dialogue: 0,0:03:23.47,0:03:26.71,Default,,0000,0000,0000,,has. Now that we've seen for\Nexamples of specific
Dialogue: 0,0:03:26.71,0:03:30.67,Default,,0000,0000,0000,,matrices, let's look at how\Nwe can write down a general
Dialogue: 0,0:03:30.67,0:03:34.27,Default,,0000,0000,0000,,matrix. Let's suppose this\Nmatrix has the symbol A and
Dialogue: 0,0:03:34.27,0:03:38.95,Default,,0000,0000,0000,,let's suppose it's got M rows\Nand N columns, so it's an M
Dialogue: 0,0:03:38.95,0:03:40.03,Default,,0000,0000,0000,,by N matrix.
Dialogue: 0,0:03:42.82,0:03:47.00,Default,,0000,0000,0000,,The number that's in the first\Nrow, first column of Matrix
Dialogue: 0,0:03:47.00,0:03:51.56,Default,,0000,0000,0000,,Capital A will write using a\Nlittle A and some subscripts 11
Dialogue: 0,0:03:51.56,0:03:56.88,Default,,0000,0000,0000,,where the first number refers to\Nthe row label and the 2nd to the
Dialogue: 0,0:03:56.88,0:04:00.30,Default,,0000,0000,0000,,column label. So this is first\Nrow first column.
Dialogue: 0,0:04:01.73,0:04:06.24,Default,,0000,0000,0000,,The second number will be a 12,\Nwhich corresponds to the first
Dialogue: 0,0:04:06.24,0:04:11.51,Default,,0000,0000,0000,,row, second column, and so on.\NThe next one will be a 1 three.
Dialogue: 0,0:04:12.50,0:04:18.47,Default,,0000,0000,0000,,And so on. Now in this matrix,\Nbecause it's an M by N matrix,
Dialogue: 0,0:04:18.47,0:04:24.02,Default,,0000,0000,0000,,it's got N columns, so the last\Nnumber in this first row will be
Dialogue: 0,0:04:24.02,0:04:26.80,Default,,0000,0000,0000,,a 1 N corresponding to 1st row.
Dialogue: 0,0:04:27.48,0:04:28.51,Default,,0000,0000,0000,,And column.
Dialogue: 0,0:04:30.99,0:04:35.64,Default,,0000,0000,0000,,What about the number in here?\NWell, it's going to be in the
Dialogue: 0,0:04:35.64,0:04:40.66,Default,,0000,0000,0000,,2nd row first column, so we'll\Ncall it a 2 one. That's the 2nd
Dialogue: 0,0:04:40.66,0:04:41.73,Default,,0000,0000,0000,,row, first column.
Dialogue: 0,0:04:42.51,0:04:45.89,Default,,0000,0000,0000,,The one here will be second\Nrow, second column.
Dialogue: 0,0:04:47.18,0:04:51.67,Default,,0000,0000,0000,,2nd row, third column, and so on\Nuntil we get to the last number
Dialogue: 0,0:04:51.67,0:04:57.22,Default,,0000,0000,0000,,in this row. Which will be a 2 N\Nwhich corresponds to 2nd row,
Dialogue: 0,0:04:57.22,0:05:01.84,Default,,0000,0000,0000,,NTH column and so on. We can\Nbuild up the matrix like this.
Dialogue: 0,0:05:01.84,0:05:06.81,Default,,0000,0000,0000,,We put all these numbers in as\Nwe want to be 'cause this matrix
Dialogue: 0,0:05:06.81,0:05:08.23,Default,,0000,0000,0000,,has got M rose.
Dialogue: 0,0:05:08.82,0:05:15.27,Default,,0000,0000,0000,,The last row here will have a\Nnumber AM one corresponding to M
Dialogue: 0,0:05:15.27,0:05:16.76,Default,,0000,0000,0000,,throw first column.
Dialogue: 0,0:05:18.23,0:05:23.16,Default,,0000,0000,0000,,NTH Row, second column and so on\Nall the way along until the last
Dialogue: 0,0:05:23.16,0:05:27.73,Default,,0000,0000,0000,,number here, which is in the M\Nthrow and the NTH column. So
Dialogue: 0,0:05:27.73,0:05:32.66,Default,,0000,0000,0000,,we'll call that a MN and that's\Nthe format in which we can write
Dialogue: 0,0:05:32.66,0:05:34.07,Default,,0000,0000,0000,,down a general matrix.
Dialogue: 0,0:05:35.08,0:05:37.18,Default,,0000,0000,0000,,Each of these numbers in the
Dialogue: 0,0:05:37.18,0:05:43.30,Default,,0000,0000,0000,,matrix. We call an element of\Nthe matrix, so A1 one is the
Dialogue: 0,0:05:43.30,0:05:48.03,Default,,0000,0000,0000,,element in the first row, first\Ncolumn. In general, the element
Dialogue: 0,0:05:48.03,0:05:54.05,Default,,0000,0000,0000,,AIJ will be the number that's in\Nthe I throw and the J column.
Dialogue: 0,0:05:55.25,0:05:58.35,Default,,0000,0000,0000,,Now some of the matrices that\Nwill come across occur so
Dialogue: 0,0:05:58.35,0:06:00.89,Default,,0000,0000,0000,,frequently or have special\Nproperties that we give them
Dialogue: 0,0:06:00.89,0:06:03.71,Default,,0000,0000,0000,,special names. Let's have a look\Nat some of those.
Dialogue: 0,0:06:05.39,0:06:08.98,Default,,0000,0000,0000,,Let's go back and look again\Nat the Matrix A. We saw a
Dialogue: 0,0:06:08.98,0:06:09.81,Default,,0000,0000,0000,,few minutes ago.
Dialogue: 0,0:06:15.32,0:06:17.60,Default,,0000,0000,0000,,This matrix has got two rows.
Dialogue: 0,0:06:18.29,0:06:20.62,Default,,0000,0000,0000,,And two columns. So it's a two
Dialogue: 0,0:06:20.62,0:06:25.39,Default,,0000,0000,0000,,by two matrix. And a matrix\Nthat's got the same number of
Dialogue: 0,0:06:25.39,0:06:29.14,Default,,0000,0000,0000,,rows and columns like this one,\Nhas we call for obvious reasons
Dialogue: 0,0:06:29.14,0:06:33.31,Default,,0000,0000,0000,,a square matrix? So this is\Nthe first example of a
Dialogue: 0,0:06:33.31,0:06:33.96,Default,,0000,0000,0000,,square matrix.
Dialogue: 0,0:06:41.64,0:06:43.89,Default,,0000,0000,0000,,Now we've already seen another\Nsquare matrix because the
Dialogue: 0,0:06:43.89,0:06:46.89,Default,,0000,0000,0000,,matrix D that we saw a few\Nminutes ago, which was this
Dialogue: 0,0:06:46.89,0:06:47.14,Default,,0000,0000,0000,,one.
Dialogue: 0,0:06:56.24,0:06:59.41,Default,,0000,0000,0000,,He's also a square matrix. This\None's got three rows.
Dialogue: 0,0:07:00.04,0:07:03.67,Default,,0000,0000,0000,,And three columns. It's a\Nthree by three matrix, and
Dialogue: 0,0:07:03.67,0:07:07.30,Default,,0000,0000,0000,,because it's got the same\Nnumber of rows and columns,
Dialogue: 0,0:07:07.30,0:07:09.12,Default,,0000,0000,0000,,that's also a square matrix.
Dialogue: 0,0:07:10.96,0:07:14.04,Default,,0000,0000,0000,,Another term I'd like to\Nintroduce is what's called a
Dialogue: 0,0:07:14.04,0:07:14.66,Default,,0000,0000,0000,,diagonal matrix.
Dialogue: 0,0:07:21.73,0:07:26.99,Default,,0000,0000,0000,,If we look again at the matrix\ND, we'll see that it has some
Dialogue: 0,0:07:26.99,0:07:30.38,Default,,0000,0000,0000,,rather special property. This\Ndiagonal, which runs from the
Dialogue: 0,0:07:30.38,0:07:34.51,Default,,0000,0000,0000,,top left to the bottom right, is\Ncalled the leading diagonal.
Dialogue: 0,0:07:41.97,0:07:45.46,Default,,0000,0000,0000,,And if you look carefully,\Nyou'll see that all the elements
Dialogue: 0,0:07:45.46,0:07:47.36,Default,,0000,0000,0000,,that are not on the leading
Dialogue: 0,0:07:47.36,0:07:52.83,Default,,0000,0000,0000,,diagonal are zeros. 0000000 A\Nmatrix for which all the
Dialogue: 0,0:07:52.83,0:07:57.93,Default,,0000,0000,0000,,elements off the leading\Ndiagonal are zero, is called
Dialogue: 0,0:07:57.93,0:07:59.62,Default,,0000,0000,0000,,a diagonal matrix.
Dialogue: 0,0:08:01.30,0:08:04.49,Default,,0000,0000,0000,,There's another special sort of\Ndiagonal matrix I'll introduce
Dialogue: 0,0:08:04.49,0:08:06.61,Default,,0000,0000,0000,,now. Let's call this one, I.
Dialogue: 0,0:08:10.41,0:08:15.86,Default,,0000,0000,0000,,Suppose this is a two by two\Nmatrix with ones on the leading
Dialogue: 0,0:08:15.86,0:08:17.53,Default,,0000,0000,0000,,diagonal and zeros everywhere
Dialogue: 0,0:08:17.53,0:08:20.34,Default,,0000,0000,0000,,else. So this is a square
Dialogue: 0,0:08:20.34,0:08:24.48,Default,,0000,0000,0000,,matrix. It's diagonal because\Neverything off the leading
Dialogue: 0,0:08:24.48,0:08:25.78,Default,,0000,0000,0000,,diagonal is 0.
Dialogue: 0,0:08:26.40,0:08:29.83,Default,,0000,0000,0000,,And it's rather special because\Non the leading diagonal, all the
Dialogue: 0,0:08:29.83,0:08:34.56,Default,,0000,0000,0000,,elements are one. Now a matrix\Nwhich has this property is
Dialogue: 0,0:08:34.56,0:08:36.02,Default,,0000,0000,0000,,called an identity matrix.
Dialogue: 0,0:08:37.58,0:08:39.38,Default,,0000,0000,0000,,Or a unit matrix?
Dialogue: 0,0:08:48.61,0:08:52.04,Default,,0000,0000,0000,,And when we're working with\Nmatrices, it's usual to reserve
Dialogue: 0,0:08:52.04,0:08:56.16,Default,,0000,0000,0000,,the letter I for an identity\Nmatrix. Now suppose we have a
Dialogue: 0,0:08:56.16,0:08:57.53,Default,,0000,0000,0000,,bigger identity matrix. Here's
Dialogue: 0,0:08:57.53,0:09:02.09,Default,,0000,0000,0000,,another one. Suppose we have a\Nthree by three identity matrix.
Dialogue: 0,0:09:05.08,0:09:07.04,Default,,0000,0000,0000,,And again, notice that\Nit's square.
Dialogue: 0,0:09:08.46,0:09:12.28,Default,,0000,0000,0000,,It's diagonal and there are ones\Nonly on the leading diagonal, so
Dialogue: 0,0:09:12.28,0:09:16.41,Default,,0000,0000,0000,,this is also an identity matrix.\NBut because this is a three by
Dialogue: 0,0:09:16.41,0:09:21.18,Default,,0000,0000,0000,,three and this ones are two by\Ntwo and we might not want to mix
Dialogue: 0,0:09:21.18,0:09:25.95,Default,,0000,0000,0000,,them up and might call this one\NI2, because this is a two by two
Dialogue: 0,0:09:25.95,0:09:30.40,Default,,0000,0000,0000,,matrix and I might call this one\NI3. But in both cases these are
Dialogue: 0,0:09:30.40,0:09:33.58,Default,,0000,0000,0000,,identity matrices and we'll see\Nthat identity matrices have a
Dialogue: 0,0:09:33.58,0:09:37.08,Default,,0000,0000,0000,,very important role to play when\Nwe look at matrix multiplication
Dialogue: 0,0:09:37.08,0:09:38.35,Default,,0000,0000,0000,,in a forthcoming video.