= Spline roughness penalty matrix
{title2=$\Gamma_{ij}=\int_a^b b_i^{\prime\prime}b_j^{\prime\prime}\,dx$}
Let $b_i=N e_i$ be the interpolating cardinal basis and set $\Gamma_{ij}=\int_a^b b_i^{\prime\prime}b_j^{\prime\prime}\,dx$. Linearity gives $J(N\mathbf v)=\mathbf v^T\Gamma\mathbf v$, and this <Gram matrix> is a <positive semidefinite matrix>. Its <null space> consists precisely of <vectors> $(\alpha+\beta x_i)_i$, by the null-space statement for the <second derivative roughness penalty>. Thus its rank is $n-2$, including the zero <matrix> when $n=2$.
Back to article page
