Welcome to a lesson
on the transpose of a matrix.
The goals of this video are
to define the transpose of a matrix,
define a symmetric matrix,
give the properties of transposed matrices,
and also to determine the transpose of a given matrix.
The transpose of an n by m matrix A
is an m by n matrix
written using this notation here, where we have
A with a superscript of capital T,
and we say 'the transpose of matrix A.'
Again, notice how the dimensions of matrix A
is n by m, the transpose matrix will have
the dimensions m by n.
And there are several ways to form
the transpose of a matrix.
One way is to write the rows of matrix A
as the columns of the transpose matrix A,
or we can write the columns of matrix A
as the rows of the transpose of matrix A.
We can also express this using this notation here,
if we focus on the subscripts here,
notice how these elements have row i, column j,
and the transpose matrix has row j, column i.
Again, we're interchanging the rows and columns.
If matrix A is equal to the transpose of matrix A,
then we say A is called a symmetric matrix.
Let's take a look at two examples
of transposing a given matrix.
I would like to start with the dimensions.
Notice matrix A is a three by two matrix,
so that means the transpose of matrix A
will be a two by three matrix.
So what I'm going to do is go through each row
in matrix A and make it the corresponding column
in the transpose of matrix A.
Since row one has elements four, negative three,
column one in the transposed matrix
will have elements four and negative three.
Row two has elements one and six,
so column two will have elements one and six.
Row three has elements negative five and two,
so column three will have elements negative five
and positive two.
We now have the transpose of matrix A.
Notice matrix B is a three by three matrix.
So the transpose of matrix B will also
be a three by three matrix.
Again, we're going to take the elements
in the rows and make them the columns
of the transposed matrix.
Row one has elements one, zero, negative two,
which means in the transposed matrix,
column one will have elements one, zero, negative two.
Row two has elements three, four, negative one.
This will form column two in the transposed matrix.
Row three forms column three in the transposed matrix,
so negative five, negative one, and positive seven.
So just to emphasize this,
notice how row one became column one,
row two became column two,
and row three became column three.
And the same thing for our first example.
Row one, column one.
Row two, column two.
And row three, column three.
Now let's talk about the properties of transposed matrices.
Property one is telling us that the transpose
of the sum of matrix A and B
is equal to the transpose of matrix A
plus the transpose of Matrix B.
Property two is telling us the transpose
of the transpose of matrix A
is just equal to matrix A.
Property three, the transpose of a scalar times matrix A
is equal to a scaler times the transpose of matrix A.
And then property four might be a little surprising.
This is telling us the transpose of matrix A
times matrix B is equal to the transpose of matrix B
times the transpose of matrix A.
Remember, matrix multiplication is not commutative,
so it is important to recognize in this property,
the order of multiplication is changing.
Property five, if A is an n by n, and invertible matrix,
then the transpose of matrix A is also invertible.
And the inverse of the transpose of matrix A
is equal to the transpose of the inverse of matrix A.
And then the last property,
matrix A times the transpose of matrix A
is equal to the transpose of matrix A
times the transpose of matrix A.
And since these are equal,
A times the transpose of A is a symmetric matrix.
Meaning if we take this product
and then find the transpose of it,
it will be the same matrix.
Let's take a look an illustration of this last property.
We're given matrix A.
We're asked to find the transpose of matrix A,
then also to find the product of matrix A and its transpose.
So notice that matrix A is a two by three matrix.
So it's transpose will be a three by two matrix.
So we'll have three rows and two columns.
So the first row in matrix A
will make up the first column of our transposed matrix.
And the second row will make up the second column
of our transposed matrix.
So we'll have five, zero, negative two,
and six, one, three.
Now we're asked to find this product,
let's go ahead and set this up.
We have a two by three and a three by two.
Since these two values are the same,
the product is possible,
and then the outer numbers tell us the dimension
of our product, so this product will be a two by two matrix.
So to find this element here in row one column one,
we multiply row one by column one.
So five times five that's 25.
Plus zero plus positive four, that's 29.
Here we're in row one, column two,
so multiply row one by column two.
So five times six, that's 30
plus zero plus negative six.
30 plus negative six is positive 24.
Here we're in row two, column one,
so multiply row two by column one.
30 plus zero plus negative six, that's 24.
And here we're in row two, column two,
so multiply row two by column two.
So we have 36 plus one, that's 37.
Plus three times three, or plus nine,
that's going to be 46.
Let's go ahead and call this matrix B.
And just to illustrate this property,
let's go ahead and find the transpose of matrix B.
So row one here will form column one,
and row two will form column two.
So we have 29, 24,
and we have 24, 46.
Notice how matrix B and the transpose of matrix B
are equal to each other.
And this will always be true.
Okay, I hope you found this helpful,
thank you for watching.