Equation (6.5), with $a$ and $t$ are vectors, K is a symmetric matrix, and $\lambda$ is real value. $$ J(a) = 0.5 a^T KK a - a^T K t + 0.5 t^T t + 0.5 \lambda a^T K a $$ The book provide the minimum solution, $$ a = (K + \lambda I)^{-1} t $$
I derived the gradient of (6.5) with respect to a as below, $$ J^{'}(a) = a^T KK - Kt + \lambda a^T K $$ and set the gradient to zero to find the minimum solution, but it does not look like the provided solution, $$ a = (KK + \lambda I)^{-1} (Kt)^T = K^-1 (K + \lambda I)^{-1} t^T K $$
In the gradient you mix row and column vectors $$J^{'}(a) = KKa - Kt + \lambda Ka =K((K+\lambda I)a-t)=0$$ If $K$ is invertible, you get $(K+\lambda I)a-t=0$.