In this video I'm going to explain what is meant by a
matrix and introduce the notation that we use when we're
working with matrices. So let's start by looking at what we mean
by a matrix and matrix is a rectangular pattern of numbers.
Let's have an example.
I'm writing down a pattern of numbers.
4 -- 113 and 9 and you see they form a rectangular pattern and
when we write down a matrix, we usually enclose the numbers with
some round brackets like that. So that's our first example of a
matrix. Let's have a look at another example which has a
different size. So suppose we have the numbers 12.
304 and again, this is a rectangular pattern of
numbers. I'll put them in round brackets like that, and
that's another example of a matrix. Let's have some more.
71
minus three to four and four.
And a final example.
A half 00.
03000 and let's say nought .7.
So here we have 4 examples of matrices.
They all have different sizes, so let's look a little bit more
about how we refer to the size of a matrix. This first matrix
here has got two rows and two columns and we describe it as a
two by two matrix. We write it as two by two like that. That's
just the size. When we write down the size of a matrix, we
always give the number of rows first and the columns second.
So this has two rows and two columns.
What about this matrix?
This is got one row.
And 1234 columns.
So this is a 1 by 4 matrix, one
row. And four columns.
This matrix has got 123 rows.
And two columns.
So it's a three by two matrix and the final example
has got three rows.
Three columns, so this is a three by three matrix, so
remember that we always give the number of rows first and
column 2nd.
The other bit of notation that we'll need is that we often use
a capital letter to denote a matrix, so we might call this
first matrix here A.
We might call the second 1B.
This one C.
And this 1D.
So there we are four examples of matrices, all of different
sizes and we now know how to describe a matrix in terms of
the number of rows and the number of columns that it
has. Now that we've seen for examples of specific
matrices, let's look at how we can write down a general
matrix. Let's suppose this matrix has the symbol A and
let's suppose it's got M rows and N columns, so it's an M
by N matrix.
The number that's in the first row, first column of Matrix
Capital A will write using a little A and some subscripts 11
where the first number refers to the row label and the 2nd to the
column label. So this is first row first column.
The second number will be a 12, which corresponds to the first
row, second column, and so on. The next one will be a 1 three.
And so on. Now in this matrix, because it's an M by N matrix,
it's got N columns, so the last number in this first row will be
a 1 N corresponding to 1st row.
And column.
What about the number in here? Well, it's going to be in the
2nd row first column, so we'll call it a 2 one. That's the 2nd
row, first column.
The one here will be second row, second column.
2nd row, third column, and so on until we get to the last number
in this row. Which will be a 2 N which corresponds to 2nd row,
NTH column and so on. We can build up the matrix like this.
We put all these numbers in as we want to be 'cause this matrix
has got M rose.
The last row here will have a number AM one corresponding to M
throw first column.
NTH Row, second column and so on all the way along until the last
number here, which is in the M throw and the NTH column. So
we'll call that a MN and that's the format in which we can write
down a general matrix.
Each of these numbers in the
matrix. We call an element of the matrix, so A1 one is the
element in the first row, first column. In general, the element
AIJ will be the number that's in the I throw and the J column.
Now some of the matrices that will come across occur so
frequently or have special properties that we give them
special names. Let's have a look at some of those.
Let's go back and look again at the Matrix A. We saw a
few minutes ago.
This matrix has got two rows.
And two columns. So it's a two
by two matrix. And a matrix that's got the same number of
rows and columns like this one, has we call for obvious reasons
a square matrix? So this is the first example of a
square matrix.
Now we've already seen another square matrix because the
matrix D that we saw a few minutes ago, which was this
one.
He's also a square matrix. This one's got three rows.
And three columns. It's a three by three matrix, and
because it's got the same number of rows and columns,
that's also a square matrix.
Another term I'd like to introduce is what's called a
diagonal matrix.
If we look again at the matrix D, we'll see that it has some
rather special property. This diagonal, which runs from the
top left to the bottom right, is called the leading diagonal.
And if you look carefully, you'll see that all the elements
that are not on the leading
diagonal are zeros. 0000000 A matrix for which all the
elements off the leading diagonal are zero, is called
a diagonal matrix.
There's another special sort of diagonal matrix I'll introduce
now. Let's call this one, I.
Suppose this is a two by two matrix with ones on the leading
diagonal and zeros everywhere
else. So this is a square
matrix. It's diagonal because everything off the leading
diagonal is 0.
And it's rather special because on the leading diagonal, all the
elements are one. Now a matrix which has this property is
called an identity matrix.
Or a unit matrix?
And when we're working with matrices, it's usual to reserve
the letter I for an identity matrix. Now suppose we have a
bigger identity matrix. Here's
another one. Suppose we have a three by three identity matrix.
And again, notice that it's square.
It's diagonal and there are ones only on the leading diagonal, so
this is also an identity matrix. But because this is a three by
three and this ones are two by two and we might not want to mix
them up and might call this one I2, because this is a two by two
matrix and I might call this one I3. But in both cases these are
identity matrices and we'll see that identity matrices have a
very important role to play when we look at matrix multiplication
in a forthcoming video.