matrix transpose example (DL)

Transpose

Getting the transpose of a matrix is really easy in NumPy. Simply access its T attribute. There is also a transpose() function which returns the same thing, but you’ll rarely see that used anywhere because typing T is so much easier. :)

For example:

m = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]])
m
# displays the following result:
# array([[ 1,  2,  3,  4],
#        [ 5,  6,  7,  8],
#        [ 9, 10, 11, 12]])

m.T
# displays the following result:
# array([[ 1,  5,  9],
#        [ 2,  6, 10],
#        [ 3,  7, 11],
#        [ 4,  8, 12]])

NumPy does this without actually moving any data in memory - it simply changes the way it indexes the original matrix - so it’s quite efficient.

However, that also means you need to be careful with how you modify objects, because they are sharing the same data. For example, with the same matrix m from above, let's make a new variable m_t that stores m's transpose. Then look what happens if we modify a value in m_t:

m_t = m.T
m_t[3][1] = 200
m_t
# displays the following result:
# array([[ 1,   5, 9],
#        [ 2,   6, 10],
#        [ 3,   7, 11],
#        [ 4, 200, 12]])

m
# displays the following result:
# array([[ 1,  2,  3,   4],
#        [ 5,  6,  7, 200],
#        [ 9, 10, 11,  12]])

Notice how it modified both the transpose and the original matrix, too! That's because they are sharing the same copy of data. So remember to consider the transpose just as a different view of your matrix, rather than a different matrix entirely.

A real use case

I don't want to get into too many details about neural networks because you haven't covered them yet, but there is one place you will almost certainly end up using a transpose, or at least thinking about it.

Let's say you have the following two matrices, called inputs and weights,

inputs = np.array([[-0.27,  0.45,  0.64, 0.31]])
inputs
# displays the following result:
# array([[-0.27,  0.45,  0.64,  0.31]])

inputs.shape
# displays the following result:
# (1, 4)

weights = np.array([[0.02, 0.001, -0.03, 0.036], \
    [0.04, -0.003, 0.025, 0.009], [0.012, -0.045, 0.28, -0.067]])

weights
# displays the following result:
# array([[ 0.02 ,  0.001, -0.03 ,  0.036],
#        [ 0.04 , -0.003,  0.025,  0.009],
#        [ 0.012, -0.045,  0.28 , -0.067]])

weights.shape
# displays the following result:
# (3, 4)

I won't go into what they're for because you'll learn about them later, but you're going to end up wanting to find the matrix product of these two matrices.

If you try it like they are now, you get an error:

np.matmul(inputs, weights)
# displays the following error:
# ValueError: shapes (1,4) and (3,4) not aligned: 4 (dim 1) != 3 (dim 0)

If you did the matrix multiplication lesson, then you've seen this error before. It's complaining of incompatible shapes because the number of columns in the left matrix, 4, does not equal the number of rows in the right matrix, 3.

So that doesn't work, but notice if you take the transpose of the weights matrix, it will:

np.matmul(inputs, weights.T)
# displays the following result:
# array([[-0.01299,  0.00664,  0.13494]])

It also works if you take the transpose of inputs instead and swap their order, like we showed in the video:

np.matmul(weights, inputs.T)
# displays the following result:
# array([[-0.01299],# 
#        [ 0.00664],
#        [ 0.13494]])

The two answers are transposes of each other, so which multiplication you use really just depends on the shape you want for the output.