[Tex/LaTex] Left align first line of equation (and other complex align)

Question

\usepackage{mathtools, amssymb, amsthm}
\begin{align}\label{eq1}
\sum_{n}\lambda^{n}(i)-\sum_{n}\lambda^{n}(j) \nonumber\\=\begin{cases}
    \lambda_{n},  &\text{if } n = i\\
    -\lambda_{n}, &\text{if } n = i\\
    0, &\text{otherwise}
    \end{cases} 
    \text{ }\forall i, j
\end{align}

First line of the equation must be left aligned (or at least adjustable, \hspace not working). \sum_{n}\lambda^{n}(i)-\sum_{n}\lambda^{n}(j) \nonumber then according to first line cases will be aligned. However, equation number should be at extreme right.

As per clarification request by @Mico The first pic from the answer.

What I need is the following pic:

The alignment of the starting of the equation is preferably left (or adjustable)

Answer 1

I'm not sure what the proposed alignment should achieve, but here it is:

\documentclass[twocolumn]{article}
\usepackage{amsmath}

\begin{document}

\begin{itemize}

\item sample equation
\begin{equation}\label{eq1}
\hspace{\dimexpr\displaywidth-\linewidth}
\begin{aligned}[b]
\sum_{n}\lambda^{n}(i)
&-\sum_{n}\lambda^{n}(j)\\
&=\begin{cases}
  \lambda_{n},  & \text{if $n = i$}\\
  -\lambda_{n}, & \text{if $n = i$}\\
  0,            & \text{otherwise}
  \end{cases} 
  \quad\forall\, i, j
\end{aligned}
\hspace{10000pt minus 1fil}
\end{equation}

\item sample equation
\begin{equation}\label{eq2}
\begin{split}
\sum_{n}\lambda^{n}(i)
&-\sum_{n}\lambda^{n}(j)\\
&=\begin{cases}
  \lambda_{n},  & \text{if $n = i$}\\
  -\lambda_{n}, & \text{if $n = i$}\\
  0,            & \text{otherwise}
  \end{cases} 
  \quad\forall\, i, j
\end{split}
\end{equation}

\end{itemize}

\end{document}