Let $ E(q)=\lambda(q-p)^T(q-p)+q^TLq $,
where $\lambda$ is a scalar, q and p are n by 1 column vectors, L is an n by n positive definite matrix (Actually Laplacian matrix, in image processing field).To minimize E(q), it's equivalent to solve :
$(L+\lambda)q=\lambda p$
I read the above in papers related to image processing. But it's difficult for me to show the equivalence among the two equations.
Would you guys please help me with the proof, or just point out reference that might help ?
To minimize $E(q)$, we require $\frac{d}{dq}E(q)=0$. Then
$$\frac{d}{dq}E(q)=\frac{d}{dq}\left[\lambda(q-p)^T(q-p)+q^TLq\right]=2\lambda(q-p)+Lq+q^T L$$ Since $L$ is positive definite and therefore symmetric, we have $q^T L = Lq$ and therefore can rearrange $\frac{d}{dq}E(q)=0$ to obtain $(L+\lambda)q=\lambda p$.