calculate precision and recall in a confusion matrix

I don't think you need summation at last. Without summation, your method is correct; it gives precision and recall for each class.

If you intend to calculate average precision and recall, then you have two options: micro and macro-average.

Read more here http://scikit-learn.org/stable/auto_examples/model_selection/plot_precision_recall.html

first, your matrix is arranged upside down. You want to arrange your labels so that true positives are set on the diagonal [(0,0),(1,1),(2,2)] this is the arrangement that you're going to find with confusion matrices generated from sklearn and other packages.

Once we have things sorted in the right direction, we can take a page from this answer and say that:

1. True Positives are on the diagonal position
2. False positives are column-wise sums. Without the diagonal
3. False negatives are row-wise sums. Without the diagonal.

\ Then we take some formulas from sklearn docs for precision and recall. And put it all into code:

import numpy as np
cm = np.array([[2,1,0], [3,4,5], [6,7,8]])
true_pos = np.diag(cm)
false_pos = np.sum(cm, axis=0) - true_pos
false_neg = np.sum(cm, axis=1) - true_pos

precision = np.sum(true_pos / (true_pos + false_pos))
recall = np.sum(true_pos / (true_pos + false_neg))

Since we remove the true positives to define false_positives/negatives only to add them back... we can simplify further by skipping a couple of steps:

 true_pos = np.diag(cm) 
 precision = np.sum(true_pos / np.sum(cm, axis=0))
 recall = np.sum(true_pos / np.sum(cm, axis=1))

For the sake of completeness for future reference, given a list of grounth (gt) and prediction (pd). The following code snippet computes confusion matrix and then calculates precision and recall.

from sklearn.metrics import confusion_matrix

gt = [1,1,2,2,1,0]
pd = [1,1,1,1,2,0]

cm = confusion_matrix(gt, pd)

#rows = gt, col = pred

#compute tp, tp_and_fn and tp_and_fp w.r.t all classes
tp_and_fn = cm.sum(1)
tp_and_fp = cm.sum(0)
tp = cm.diagonal()

precision = tp / tp_and_fp
recall = tp / tp_and_fn