Exercise 8.5 - Symmetric version of l2 regularized multinomial logistic regression

We borrow the results from exercise 8.5. Once the $l_2$ prior is introduced, the likelihood becomes:

Therefore (8.126) becomes:

At the unique optimum we have $\forall <!--\; FUNCTION\; APPLICATION\; -->c$,

which is identical to:

Therefore:

whose value is zero since ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_c{y}_{\mathit{𝑖𝑐}}={\sum }_c{\mu }_{\mathit{𝑖𝑐}}=1$.