# tf.multiply vs tf.matmul to calculate the dot product

## Issue

I have a matrix (of vectors) X with shape [3,4], and I want to calculate the dot product between each pair of vectors (X[1].X[1]) and (X[1].X[2])…etc.

I saw a cosine similarity code were they use

tf.reduce_sum(tf.multyply(X, X),axis=1)

to calculate the dot product between the vectors in a matrix of vectors.However, this result in only calculates the dot product between (X[i], X[i]).

I used tf.matmul(X, X, transpose_b=True) which calculate the dot product between every two vectors but I am still confused why tf.multiply didn’t do this I think the problem with my code.

the code is:

``````data=[[1.0,2.0,4.0,5.0],[0.0,6.0,7.0,8.0],[8.0,1.0,1.0,1.0]]
X=tf.constant(data)
matResult=tf.matmul(X, X, transpose_b=True)

multiplyResult=tf.reduce_sum(tf.multiply(X,X),axis=1)
with tf.Session() as sess:
print('matResult')
print(sess.run([matResult]))
print()
print('multiplyResult')
print(sess.run([multiplyResult]))
``````

The output is:

``````matResult
[array([[  46.,   80.,   19.],
[  80.,  149.,   21.],
[  19.,   21.,   67.]], dtype=float32)]

multiplyResult
[array([  46.,  149.,   67.], dtype=float32)]
``````

## Solution

`tf.multiply(X, Y)` or the `*` opperator does element-wise multiplication so that

``````[[1 2]    [[1 3]      [[1 6]
[3 4]] .  [2 1]]  =   [6 4]]
``````

wheras `tf.matmul` does matrix multiplication so that

``````[[1 0]    [[1 3]      [[1 3]
[0 1]] .  [2 1]]  =   [2 1]]
``````

using `tf.matmul(X, X, transpose_b=True)` means that you are calculating `X . X^T` where `^T` indicates the transposing of the matrix and `.` is the matrix multiplication.

`tf.reduce_sum(_, axis=1)` takes the sum along 1st axis (starting counting with 0) which means you are suming the rows:

``````tf.reduce_sum([[a b], [c, d]], axis=1) = [a+b, c+d]
``````

This means that:

``````tf.reduce_sum(tf.multiply(X, X), axis=1) = [X[1].X[1], ..., X[n].X[n]]
``````

so that is the one you want if you only want the norms of each rows. On the other hand

`````` tf.matmul(X, X, transpose_b=True) = [[ X[1].X[1], X[1].X[2], ..., X[1].X[n]],
[X[2].X[1], ..., X[2].X[n]],
...
[X[n].X[1], ..., X[n].X[n]]
``````

so that is what you need if you want the similarity between all pairs of rows.