(This question pertains to a larger model proposed in this SIGRAPH paper, but I've pulled out the pertinent question and generalized it.)
Let's say I have two large matrices $M$ and $L \in\mathbb{R}^{3n \times 3n}$, where $M$ is some blocked diagonal mass matrix such that $M = diag[m_1I_3, m_2I_3,...,m_nI_3]$ where $m_i>0$, and $L$ is the Kronecker product of some Laplacian matrix $L_q \in \mathbb{R}^{n \times n}$ and $I_3$ ($L = L_q \otimes I_3)$.
I'm trying to compute the Cholesky factorization of $M + aL$, where $a \in \mathbb{R}, a > 0$. The paper that I found this problem in claims that since $L$ is symmetric positive semi-definite and $M$ is symmetric positive-definite, $M + aL$ is guaranteed to be positive-definite, but when I attempt to compute the Cholesky of the matrix the LA library I'm using (Math.Net Numerics) says the matrix isn't positive-definite.
The matrix in question is too large to check practically by hand (700x700-ish) and I'm a little iffy on the theory here. Is the paper incorrect about $M + aL$ being positive-definite, or have I made a mistake in my code?
The matrix is positive-definite: if $x \not= 0$ then $$x^T (M+aL) x = x^T M x + a x^TLx > 0,$$ since $x^TMx > 0$ and $x^TLx \geq 0$.