Here’s some part Appendix C “optimal step sizes” of[1]:
$\theta_{t+1}=\theta_t-\alpha H(\theta_t)^{-1}g(\theta_t)-----------(1)$θt+1​=θt​−αH(θt​)−1g(θt​)−−−−−−−−−−−(1)
$\alpha:\text{learning rate,also called step size in [1]}$α:learning rate,also called step size in [1]

Quotation from[1]:“if we project the gradient in the basis of eigenvectors,we get:”
$g(\theta)=\Sigma_{i=1}^N[g(\theta)^Tv_i]v_i-----------(2)$g(θ)=Σi=1N​[g(θ)Tvi​]vi​−−−−−−−−−−−(2)
I can not understand (2) ,so I design a simple example:

let $g(\theta)=\left\{ \begin{matrix} 2 & 3 \\ 1 & 2 \end{matrix} \right\}$g(θ)={21​32​}

then its eigen values and eigen vectors are :
$\lambda_{1}=2+\sqrt{3},v_1=(\sqrt{3},1)$λ1​=2+3​,v1​=(3​,1)

$\lambda_2=2-\sqrt{3},v_2 =(-\sqrt{3},1)$λ2​=2−3​,v2​=(−3​,1)

$g^T(\theta)=\left\{ \begin{matrix} 2 & 1\\ 3 & 2 \end{matrix} \right\}_{2·2}$gT(θ)={23​12​}2⋅2​

the dimension of $g^T(\theta)$gT(θ)is 2x2,which is NOT be compatible with $v_1$v1​and $v_2$v2​,so
$[g^T(\theta)v_i]v_i$[gT(θ)vi​]vi​ in (2) can NOT be computed,

Could you tell me where am I wrong?
Thanks~~!

Reference:
[1]Negative eigenvalues of the hssian in deep neural networks