https:/.../symmetricmatricesf61mb-aspect.mp4

In this video I'm going to
explain what's meant by a

symmetric matrix and the
transpose of a matrix. Let's

have a look at a matrix.

This one I'm going to call M and
it's the Matrix 4.

Minus 1 -- 1 nine.

What I'd like to do is focus on
the leading diagonal. Remember,

the leading diagonal is the one
from top left to bottom right.

That's this one here.

And if we imagine that this is a
mirror, you'll see that there's

a mirror image across across
this leading diagonal of these

elements. The element minus one.
Here is the same as that here,

and a matrix with that property
is called a symmetric matrix.

Let's have a look at a
slightly larger one.

This one will be a three by
three matrix. Let's suppose it's

got the elements 273794.

347 and again focus on the
leading diagonal here.

And look across the leading
diagonal at the particular

elements. We've got a 7 and A7.

Three and three and four and
four, so this leading diagonal

acts a little bit like a
mirror line. It's a line of

symmetry, so both this matrix
N and the previous one M are

called symmetric matrices.

One thing I'd like to do now is
introduce what's called the

transpose of the matrix. If we
take any matrix A, for example.

So let's go with the Matrix A
from before 4 -- 113 nine.

If we look at the first row 4
-- 1 and we form a new matrix

where the first column is,
this row 4 -- 1, so the first

column is 4 -- 1.

And this row here 13 nine
becomes the second column

here 13 nine we say that
this new matrix here is

obtained by taking the
transpose of the original

matrix, and we call this.

The transpose matrix and we
denote it by a with a

superscript T for transpose,
so this matrix A transpose is

the transpose of this. It's
obtained by interchanging the

rows and columns. So the
first row becomes the first

column, the 2nd row becomes
the second column.

If we look at the matrix M We
started with here and try and

find the transpose of it. Let's
do that up here.

The transpose of this matrix
is obtained by interchanging

the rows and columns. So the
first row 4 -- 1 becomes the

first column.

And the 2nd row minus 19 becomes
the second column you'll see in

this particular case that the
matrix M and the matrix M

transpose are the same. So if
you have a symmetric matrix,

it's the same as its transpose.
The same will be true of matrix

N here, which you can verify for
yourself. So symmetric matrix is

actually one which is which has
the property that a is equal to

its transpose. And that's

another definition. Of what we
mean by a symmetric matrix.

Let's have a look at another
example of finding the transpose

of a matrix. We can find the
transpose of any matrix. It

doesn't have to be a square
matrix. Let's have a look at an

example such as finding the
transpose of matrix C, which was

seven 1 -- 3, two, 4, four.

Note that this is a three row
two column matrix.

When we find it's transpose,
what we do is we take the first

row. And it becomes the first
column in the transpose matrix.

The 2nd row becomes the second
column and the final Row 4, four

becomes the final column. So
what we've done is we've

interchanged the rows and the
columns to form the transpose,

and this one we would denote as
C with a superscript T for

transpose. And note that in this
case the resulting matrix now

has got two rows and three
columns, so this is a two by

three matrix and this results is
true in general as well.

We find a transpose by
interchanging the rows and

columns. You'll find that, say,
three by two becomes a 2 by 3.

Four by three will become a 3 by
4 and M by N would become an N

by M and so on.