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.