So I am trying to study Isomaps for non-linear dimensionality reduction, and I cannot understand how we reach the formula for C as : $$C = − \frac{1}{2m}HD^{2}H$$ So we have the centering matrix, $$H = I- \frac{1}{m} 11^{T}$$
We start by multiplying it to either sides of $D^{2}$ in the distance equation. $$D^{2} = a1^{T} + 1^{T} − 2^{T}$$ where a = $[(z^{1})^{T}z^{1}, (z^{2})^{T}z^{2}, ... ..,(z^{m})^{T}z^{m}]$.
Multiplying H on either sides of $D^{2}$, I understand how $a1^{T}H=0.$ But where does the -$\frac{1}{2m}$ come from in C? Is there anything I'm missing? Thankyou so much in advance!