Vandermonde determinant

The Vandermonde product ${\prod }_{i<j}(r_i-r_j)$ is alternating, and its square is the discriminant of the monic polynomial with roots $r_i$.

For a nonnegative weight $w$ and monic polynomials $p_j$ of degree $j$, this determinant is ${\prod }_i\sqrt{w(x_i)}{\prod }_{i<j}(x_j-x_i)$. Its square is the product of the weights times the squared Vandermonde determinant. For normalized weighted polynomials it differs by the product of their norm factors.

Evaluating arbitrary monic polynomials of successive degrees $0,\dots ,m-1$ gives the same determinant as evaluating the monomials of those degrees. The basis-change matrix is triangular with diagonal entries one, so its determinant is one. This permits an orthogonal polynomial basis to replace the monomials in eigenvalue densities.

The left side is an alternating polynomial of total degree one more than $\Delta (x)$. Dividing by the Vandermonde determinant gives a symmetric homogeneous polynomial of degree one, necessarily $a{\sum }_i{x}_i+bt$. At $t=0$, $a=1$; differentiation in $t$ and the Euler theorem for homogeneous functions give $b=\left(\genfrac{}{}{0ex}{}{m}{2}\right)$. Thus the identity holds as a polynomial identity, even at repeated coordinates. At shifted partition coordinates with $t=-1$, it proves the removable-corner recurrence in the hook-length formula.

For distinct ${\ell }_i$ and $c_j$, expand the exponential entries in powers of $t$. Nonzero determinant terms first occur at the distinct powers $0,1,\dots ,m-1$. The coefficient is the product of their two Vandermonde determinants divided by the factorial product. Ratios of such alternants yield the Weyl dimension formula by taking a torus character to the identity.

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 $r$.