Understanding how TensorFlow works with matrices is very important in understanding the flow of data through computational graphs.

Many algorithms depend on matrix operations. TensorFlow gives us easy-to-use operations to perform such matrix calculations. For all of the following examples, we first create a graph session by running the following code:

import tensorflow as tf 
sess = tf.Session() 

We will proceed with the recipe as follows:

1. Creating matrices: We can create two-dimensional matrices from NumPy arrays or nested lists, as we described in the Creating and using tensors recipe. We can also use the tensor creation functions and specify a two-dimensional shape for functions such as zeros(), ones(), truncated_normal(), and so on. TensorFlow also allows us to create a diagonal matrix from a one-dimensional array or list with the diag() function, as follows:

identity_matrix = tf.diag([1.0, 1.0, 1.0]) 
A = tf.truncated_normal([2, 3]) 
B = tf.fill([2,3], 5.0) 
C = tf.random_uniform([3,2]) 
D = tf.convert_to_tensor(np.array([[1., 2., 3.],[-3., -7., -1.],[0., 5., -2.]])) 
print(sess.run(identity_matrix)) 
[[ 1.  0.  0.] 
 [ 0.  1.  0.] 
 [ 0.  0.  1.]] 
print(sess.run(A)) 
[[ 0.96751703  0.11397751 -0.3438891 ] 
 [-0.10132604 -0.8432678   0.29810596]] 
print(sess.run(B)) 
[[ 5.  5.  5.] 
 [ 5.  5.  5.]] 
print(sess.run(C)) 
[[ 0.33184157  0.08907614] 
 [ 0.53189191  0.67605299] 
 [ 0.95889051 0.67061249]] 
print(sess.run(D)) 
[[ 1.  2.  3.] 
 [-3. -7. -1.] 
 [ 0.  5. -2.]] 

2. Addition, subtraction, and multiplication: To add, subtract, or multiply matrices of the same dimension, TensorFlow uses the following function:

print(sess.run(A+B)) 
[[ 4.61596632  5.39771316  4.4325695 ] 
 [ 3.26702736  5.14477345  4.98265553]] 
print(sess.run(B-B)) 
[[ 0.  0.  0.] 
 [ 0.  0.  0.]] 
Multiplication 
print(sess.run(tf.matmul(B, identity_matrix))) 
[[ 5.  5.  5.] 
 [ 5.  5.  5.]] 

It is important to note that the matmul() function has arguments that specify whether or not to transpose the arguments before multiplication or whether each matrix is sparse.

3. The transpose: Transpose a matrix (flip the columns and rows) as follows:

print(sess.run(tf.transpose(C))) 
[[ 0.67124544  0.26766731  0.99068872] 
 [ 0.25006068  0.86560275  0.58411312]] 

Again, it is worth mentioning that reinitializing gives us different values than before.

4. Determinant: To calculate the determinant, use the following:

print(sess.run(tf.matrix_determinant(D))) 
-38.0 

5. Inverse: To find the inverse of a square matrix, see the following:

print(sess.run(tf.matrix_inverse(D))) 
[[-0.5        -0.5        -0.5       ] 
 [ 0.15789474  0.05263158  0.21052632] 
 [ 0.39473684  0.13157895  0.02631579]] 

6. Decompositions: For the Cholesky decomposition, use the following:

print(sess.run(tf.cholesky(identity_matrix))) 
[[ 1.  0.  1.] 
 [ 0.  1.  0.] 
 [ 0.  0.  1.]] 

7. Eigenvalues and eigenvectors: For eigenvalues and eigenvectors, use the following code:

print(sess.run(tf.self_adjoint_eig(D)) 
[[-10.65907521  -0.22750691   2.88658212] 
 [  0.21749542   0.63250104  -0.74339638] 
 [  0.84526515   0.2587998    0.46749277] 
 [ -0.4880805    0.73004459   0.47834331]] 

Note that the self_adjoint_eig() function outputs the eigenvalues in the first row and the subsequent vectors in the remaining vectors. In mathematics, this is known as the eigendecomposition of a matrix.

TensorFlow provides all the tools for us to get started with numerical computations and adding such computations to our graphs. This notation might seem quite heavy for simple matrix operations. Remember that we are adding these operations to the graph and telling TensorFlow which tensors to run through those operations. While this might seem verbose now, it helps us understand the notation in later chapters when this way of computation will make it easier to accomplish our goals.