Is this algebraic identity obvious? $\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1$
Solution 1:
It's the Lagrange interpolation polynomial for the constant function $1$ evaluated at $0$: $$\sum_{i=1}^n 1 \prod_{j\neq i} {{\lambda_j-0}\over \lambda_j-\lambda_i}=1$$
In general, you have that the polynomial below interpolates the data points $(\lambda_i,y_i$): $$\sum_{i=1}^n y_i \prod_{j\neq i} {{\lambda_j-x}\over \lambda_j-\lambda_i}$$
See http://en.wikipedia.org/wiki/Lagrange_polynomial