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

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In this video I'm going to
explain what's meant by a
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symmetric matrix and the
transpose of a matrix. Let's
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have a look at a matrix.
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This one I'm going to call M and
it's the Matrix 4.
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Minus 1 -- 1 nine.
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What I'd like to do is focus on
the leading diagonal. Remember,
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the leading diagonal is the one
from top left to bottom right.
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That's this one here.
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And if we imagine that this is a
mirror, you'll see that there's
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a mirror image across across
this leading diagonal of these
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elements. The element minus one.
Here is the same as that here,
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and a matrix with that property
is called a symmetric matrix.
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Let's have a look at a
slightly larger one.
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This one will be a three by
three matrix. Let's suppose it's
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got the elements 273794.
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347 and again focus on the
leading diagonal here.
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And look across the leading
diagonal at the particular
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elements. We've got a 7 and A7.
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Three and three and four and
four, so this leading diagonal
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acts a little bit like a
mirror line. It's a line of
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symmetry, so both this matrix
N and the previous one M are
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called symmetric matrices.
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One thing I'd like to do now is
introduce what's called the
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transpose of the matrix. If we
take any matrix A, for example.
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So let's go with the Matrix A
from before 4 -- 113 nine.
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If we look at the first row 4
-- 1 and we form a new matrix
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where the first column is,
this row 4 -- 1, so the first
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column is 4 -- 1.
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And this row here 13 nine
becomes the second column
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here 13 nine we say that
this new matrix here is
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obtained by taking the
transpose of the original
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matrix, and we call this.
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The transpose matrix and we
denote it by a with a
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superscript T for transpose,
so this matrix A transpose is
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the transpose of this. It's
obtained by interchanging the
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rows and columns. So the
first row becomes the first
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column, the 2nd row becomes
the second column.
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If we look at the matrix M We
started with here and try and
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find the transpose of it. Let's
do that up here.
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The transpose of this matrix
is obtained by interchanging
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the rows and columns. So the
first row 4 -- 1 becomes the
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first column.
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And the 2nd row minus 19 becomes
the second column you'll see in
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this particular case that the
matrix M and the matrix M
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transpose are the same. So if
you have a symmetric matrix,
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it's the same as its transpose.
The same will be true of matrix
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N here, which you can verify for
yourself. So symmetric matrix is
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actually one which is which has
the property that a is equal to
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its transpose. And that's
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another definition. Of what we
mean by a symmetric matrix.
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Let's have a look at another
example of finding the transpose
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of a matrix. We can find the
transpose of any matrix. It
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doesn't have to be a square
matrix. Let's have a look at an
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example such as finding the
transpose of matrix C, which was
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seven 1 -- 3, two, 4, four.
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Note that this is a three row
two column matrix.
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When we find it's transpose,
what we do is we take the first
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row. And it becomes the first
column in the transpose matrix.
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The 2nd row becomes the second
column and the final Row 4, four
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becomes the final column. So
what we've done is we've
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interchanged the rows and the
columns to form the transpose,
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and this one we would denote as
C with a superscript T for
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transpose. And note that in this
case the resulting matrix now
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has got two rows and three
columns, so this is a two by
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three matrix and this results is
true in general as well.
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We find a transpose by
interchanging the rows and
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columns. You'll find that, say,
three by two becomes a 2 by 3.
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Four by three will become a 3 by
4 and M by N would become an N
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by M and so on.
Title:
https:/.../symmetricmatricesf61mb-aspect.mp4
Video Language:
English
Duration:
05:03

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