def f_c_v2(M):
    return M / M.sum(1).reshape((M.shape[0], 1))

1. Row Summation
   • M.sum(1) calculates the sum along axis 1 (the “horizontal” direction in a 2D array), which means it sums each row individually.
   • If M has shape $(n,m)$, then M.sum(1) produces a 1D array of length n (one sum per row).
2. Reshaping for Broadcasting
   • The result of M.sum(1) is a 1D array of shape $(n,)$.
   • Calling .reshape((M.shape[0], 1)) converts that 1D array into a 2D array of shape $(n,1)$.
   • This step is crucial for broadcasting: when we divide M (shape $(n,m)$) by a $(n,1)$ array, numpy automatically applies the appropriate row sum to each element in that row.
3. Element-wise Division
   • M / [reshaped row sums] performs an element-wise division, matching each row’s elements with its corresponding row sum.
   • That is, each element in row $i$ of M gets divided by M.sum(1)[i].
4. Return the Normalised Matrix
   • Because each row is divided by its own sum, the new matrix has rows that each add up to 1.
   • This operation creates and returns a new array; it does not overwrite the original M.

Why This is Compact

• Instead of writing loops, this uses numpy’s vectorized function calls.
• Summing over rows (sum(1)) and reshaping the resulting array ensures easy broadcasting over columns.
• The division is done in-place in a single statement: M / ….

Hence, this function succinctly normalises each row of the matrix to sum to 1, showcasing a classic and powerful use of numpy broadcasting.