Tensorflow, Keras: How to create a trainable variable that only update in specific positions?

You have two different tools to address this problem.

1. You can create the variables you need and rearrange them into the desired form.
2. You can create more variables than you need then discard some to reach the desired form.

Both approach are not exclusive and you could you a mix of successives steps of type #1 and #2.

For example, for your first example (diagonal matrix), we can use approach #1.

w = tf.Variable(tf.zeros(n))
A = tf.diag(w) # creates a diagonal matrix with elements of w

For your second, more complex example, we could use approach #2.

A = tf.Variable(tf.zeros((n, n)))
A = tf.matrix_band_part(A, 1, 1) # keep only the central band of width 3
A = tf.matrix_set_diag(A, tf.ones(n)) # set diagonal to 1