Vandermonde matrices nullspaces and notation

I'm following the book Mimetic Discretization Methods by Castillo and Miranda, reading the page 209-211, it states the following:

Let $V(m;G)$ be a Vandermonde matrix, of order $m$ with generator $G = \{ g_1, g_2, \ldots, g_n \}$. Where the order implies the highest power of Vandermonde matrix, therefore $V(m;G)$ has $(m+1)$ rows and $n$ columns.

Let \begin{align} G_1 &= \{ -1/2, 1/2, 3/2, 5/2, 7/2, 9/2 \} \\ G_2 &= \{ -3/2, -1/2, 1/2, 3/2, 5/2, 7/2 \} \\ G_3 &= \{ -5/2, -3/2, -1/2, 1/2, 3/2, 5/2 \} \\ G_4 &= \{ -7/2, -5/2, -3/2, -1/2, 1/2, 3/2 \} \end{align}

Generators of the following matrices

\begin{align} V_1 &=V(4;G_1) \\ V_2 &=V(4;G_2) \\ V_3 &= V(4;G_3) \\ V_4 &= V(4;G_4) \end{align}

Just for sake of example $V_1$ looks like:

\begin{align} V_1 = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 \\ -1/2 & 1/2 & 3/2 & 5/2 & 7/2 & 9/2 \\ (-1/2)^2 & (1/2)^2 & (3/2)^2 & (5/2)^2 & (7/2)^2 & (9/2)^2 \\ (-1/2)^3 & (1/2)^3 & (3/2)^3 & (5/2)^3 & (7/2)^3 & (9/2)^3 \\ (-1/2)^4 & (1/2)^4 & (3/2)^4 & (5/2)^4 & (7/2)^4 & (9/2)^4 \\ \end{bmatrix} \end{align}

The null space of this matrix is given by $Ker(V_1) =[-1 \; 5 \; -10\; 10\; -5\; 1]^T$, that is easily verifiable just by $V_1 \cdot Ker(V_1)$ that results in the null vector, as expected.

The other Vandermonde matrices $V_2, V_3, V_4$ share the same null space. The book then states:

In general, the null space of a $k$-th order Vandermonde matrix has dimension $\frac{1}{2}k -1$.

My first question: There is no references or proofs to support this claim. Is this true? How can I prove that? What are conditions to that be true. I could not find anywhere this affirmation.

The text continues, and we need to solve the following system

\begin{equation} V_i a_i^T = b \end{equation}

Where $V_i$ are the Vandermonde matrices above, $a_i$ is a unknown vector and $b = [ 0 \;1 \;0 \;0\; 0 ]$

Lets only focus in $i=1$ since the construction to other matrices are equivalent. We then need to solve the following:

\begin{equation} V_1 a_1^T = b \end{equation}

completely written as

\begin{equation} \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 \\ -1/2 & 1/2 & 3/2 & 5/2 & 7/2 & 9/2 \\ (-1/2)^2 & (1/2)^2 & (3/2)^2 & (5/2)^2 & (7/2)^2 & (9/2)^2 \\ (-1/2)^3 & (1/2)^3 & (3/2)^3 & (5/2)^3 & (7/2)^3 & (9/2)^3 \\ (-1/2)^4 & (1/2)^4 & (3/2)^4 & (5/2)^4 & (7/2)^4 & (9/2)^4 \\ \end{bmatrix}\begin{bmatrix} a_{11} \\a_{12} \\a_{13} \\a_{14} \\a_{15} \\a_{16} \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \end{equation}

Then the text declares that

\begin{align} a_1 = \left[ -\frac{11}{12} \, \frac{17}{24} \, \frac{3}{8} \, -\frac{5}{24} \, \frac{1}{24} \, 0 \right] + \alpha_1 Ker(V_1) \end{align}

for some $\alpha_1 \in \mathbb{R}$.

My second question is: where that vector in first term came from? Is that true? How can I show that?

I'm able to solve that system using least squares, but could not find a correlation.

Some tries: (EDIT)

Definition (Null Space): $Ker(V)$ is a null space of $V$ if an only if $V \cdot \alpha Ker(V) = \vec{0}$ for any $\alpha \in \mathbb{R}$.

Lets say by hypothesys that

\begin{align} a &= x + \alpha Ker(V) \end{align}

Since we are looking for (I will drop the indexes for sake of notation and typing):

\begin{align} V \cdot a &= b \\ a &= V^{-1} b \end{align} by the hypothesis \begin{align} x + \alpha Ker(V) &= V^{-1} b \\ Vx + V \left( \alpha Ker(V) \right) &= b \end{align} by Null Space definition \begin{align} Vx &= b \end{align}

We are just back where we started, seems a tautology in my view.

Anyway I also know that

\begin{align} a &= x + \alpha Ker(V) \\ x &= a - \alpha Ker(V) \end{align}

I can find $a$ by least squares, and have $Ker(V)$ by the null space analysis. Therefore I just need to find $\alpha$ to uniquely identify $x$.

• Are you sure it's stated as “the null space of an $m$-th order Vandermonde matrix has dimension $\dfrac m2-1$”? For $n=|G|=m+1$, $V(m; G)$ is invertible and thus the dimension is $0$.
– Ѕааԁ
Jun 23 '18 at 7:36
• Also $\frac{m}{2}-1$ is not necessarily an integer...also en.wikipedia.org/wiki/Vandermonde_matrix#Resulting_properties Jun 25 '18 at 20:26

EDIT. We consider $V(m,G)\in M_{m+1,n}$ where we assume that $n\geq m+1$.

i) Thus $V(m,G):\mathbb{R}^n\rightarrow \mathbb{R}^{m+1}$. The first $m+1$ columns of $V$ realize the transpose of a standard square Vandermonde matrix $W$ which is invertible.

Thus $rank(V)=m+1$ ($V$ is surjective) and $dim(\ker(V))=n-m-1$.

In your example, $n=3m/2$, that implies $dim(\ker(V))=m/2-1$.

ii) We want to solve the equation in $x$, $Vx=e_2$. Since $V$ is surjective, there is particular solution $x_0$. If $x$ is the general solution, then $V(x_0)=V(x)$ $\iff$ $V(x-x_0)=0$ $\iff$ $x-x_0\in \ker(V)$ $\iff$ $x\in x_0+\ker(V)$. Finally, the set of solutions is $x_0+\ker(V)$.

We search $x_0$ in the form $[p,q,r,s,t,0]^T$. Then the equation to solve is equivalent to $W[p,q,r,s,t]^T=e_2$, where $W$ is the above invertible matrix. You can solve this last equation by Gauss method and obtain the unique solution $[-11/12,17/24,3/8,-5/24,1/24]^T$.

• Loup, can you point any reference or proof about invertibility of any Vandermonde matrix of n-th order?
– Lin
Jun 25 '18 at 10:20
• Loup about ii, can you please elaborate the following statement in ttfont: "If $Vx=Vy$, then $x−y \in Ker(V)$ and the set of solutions is x=x0+ker(V) where x0 is a particular solution". That is exactly where I'm struggling, I'm probably missing some definition or theorem.
– Lin
Jun 25 '18 at 10:34
• cf. my edit in my answer.
– user91684
Jun 25 '18 at 20:21