Transpose of a Matrix

In this lesson we will discuss the transpose of a matrix.

Let’s start from vector’s we already know that vectors are either rows or columns.

So let x be the column vector.

One two three.

There is a special operation which can turn it into a row vector.

We can transpose it transposing the column vector one to three would yield in a row vector 1 to 3.

We denote this operation with the letter T as a superscript Therefore X transposed or the column vector

1 2 3 transposed equals the row vector 1 2 3.

Let’s see another one the row vector one two three transposed equals the column vector 1 two 3.

So this is a pretty neat operation if we transpose x twice we get x.

How about another example the row vector 2 minus 2 0 5 when transposed becomes the column vector 2 minus

2 0 5 very easy right.

However there are several very important considerations here.

First when we transpose a vector we are not losing any information.

The values are not changing or transforming only their position is second transposing the same vector

twice yields the initial vector third our initial vector 1 to 3 had a length of three.

In other words it was three by one matrix by transposing it remained a vector of length 3 but it became

a 1 by 3 matrix.

This last consideration gives a peek into how transposing works for matrices.

It turns all of its rows into columns and vice versa in terms of dimensions when transposed an M by

and matrix becomes an end by M matrix.

Let’s check out an example shall we.

Here is the matrix.

A live 12 on the first row and mine is 3 0.

14 on the second row.

A transposed equals five minus three on the first row 12 0 on the second and 6 14 on the third a is

a two by three Matrix a transposed is a three by two Matrix.

We can see that the first column of A became the first row a transposed the second column of A became

the second row of a transposed and the third column became the third row.

This is how we transpose the matrix the initial matrix A is nothing more than a collection of two row

vectors five 12 12:6 and minus 3 0 14 transposing the matrix A means transposing each of those vectors

and putting them back together.

So transposing matrices is nothing more than transposing a bunch of vectors.

All right let’s get in Jupiter and see how this works out there.

I’ll first declare a as an NPR-A in order to get the transpose of A.

We simply write a dot capital T.

That’s all great.

We will see two more examples just to solidify the knowledge taking the two by two matrix B containing

5 3 minus 2 4 and transposing it.

We get another two by two Matrix this time.

Five minus two three four

OK.

Finally this four by 2 matrix C when transposed equals that one again.

Notice that this operation is nothing more than transposing several vectors one by one right here are

some further clarifications any scalar when transposed equals itself.

Scalars are not the most interesting concepts you see.

So let’s move on to vectors.

We have the vector x equal to 1 2 3 transposing it would yield the same result.

That’s because in Python one dimensional arrays don’t really get transposed.

Remember a shape it is three comma.

If I reshape it into a three by one matrix or a two dimensional array I will be able to transpose it

though that’s another peculiarity you must consider when working with any arrays in Python OK.

Great.

In the next lecture we will explore multiplication.

Thanks for watching.