# False positives and False negatives of three or more classes

I have a $10\times 10$ confusion matrix $M$ generated after to execute an KNN classification process for digits recognition (0,1,2...9).

As usual, each row of $M$ represent the "true/real" class of the digit and the columns represent the "assigned/predicted" class given by KNN.

In that way $M_{ij}$ represent the total quantity of the "digits $i$" classified as a "digit $j$". In particular, when $i=j$ (diagonal elements) $M_{ii}$ have the total quantity of the "digits $i$" correctly classifieds as "digits $i$". In all the other inputs of $M$ we have the classification failures.

For example, in the input $M_{1,2}$ we have the total of true $1$'s wrongly classifieds as a number $2$. In the input $M_{3,8}$ we have the total of true $3$'s wrongly classifieds as a number $8$, and so on.

I the case where exist only two classes, is common to calculate the ratio of false positives and the ratio of false negatives.

So my question is, how to calculate the equivalent rations for three or more classes, in this case, $10$ classes.