Vandermonde matrix ($V_{ij}={\lambda }_i^{j-1}$)

A Vandermonde matrix has entries $V_{ij}={\lambda }_i^{j-1}$. Distinct nodes give determinant ${\prod }_{i<j}({\lambda }_j-{\lambda }_i)\ne 0$, explaining independence of polynomial evaluations and distinct-eigenvalue Krylov subspaces.

A generalized Vandermonde matrix evaluates a chosen list of functions ${\varphi }_0,\dots ,{\varphi }_n$ at interpolation nodes $x_0,\dots ,x_n$. For a basis of polynomials of polynomial degree at most $n$ and distinct nodes, it is an invertible matrix by uniqueness of polynomial interpolation. Choosing monomials recovers the ordinary Vandermonde matrix.

Collect the distinct nonzero values among the ${\lambda }_j$ as ${\mu }_1,\dots ,{\mu }_r$ with positive multiplicities $m_j$. The first $r$ power sums imply ${\sum }_j{m}_j{\mu }_j^k=0$ for $1\le k\le r$. The coefficient matrix is an invertible weighted Vandermonde matrix, so all $m_j$ vanish, a contradiction unless $r=0$. Its invertibility also follows because a polynomial ${\sum }_{k=1}^r{a}_k{z}^k$ vanishing at zero and all the ${\mu }_j$ has $r+1$ distinct roots and degree at most $r$.