The Lee-carter model aims to model central mortality rates using the following
$$
log(m_{x,t}) = a_{x} + b_{x}k_{t} + \epsilon_{x,t}
$$
where $\epsilon_{x,t} \sim N(0,\sigma^{2})$
The following conditions are also usually imposed
$$
\sum_{t}k_{t} = 0
$$
$$
\sum_{x}b_{x} = 1
$$
These conditions imply that the $a_{x}$ values are simply the averages over time of the $Log(m_{x,t})$. The model cannot be fitted by regression methods as there
are no known covariates so Lee and Carter used SVD to derive an ordinary least squares solution.
In this paper - A Review of the Lee-Carter Method, it says that after subtracting $a_{x}$ from each column of $log(m_{x,t})$ the first term of the SVD is used to derive $b_{x}$ and $k_{t}$
Specifically, $b_{x}$ is obtained from the first left singular vector (adjusting for the $b_{x}$ constraint), and $k_{t}$ is the product of the leading singular value, the first right singular vector and the sum of the first left singular vector.
I don't quite understand how the formulations of $b_{x}$ and $k_{t}$ are arrived at, could somebody please provide details? why are only the first singular vector/values used? why is $b_{x}$ related to just the left singular vector and $k_{t}$ involves both and not the other way around? The original paper by Lee and Carter also skips over the details of this part.
Thanks in advance for any help!