-
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.