Triangular Orthonormalization of a Positive Definite Gram Matrix

Let $d\ge1$ be a natural number and let $G$ be a real $d\times d$ matrix that is symmetric, $G^{\top}=G$ with the transpose, and positive definite: for every nonzero $c\in\mathbb{R}^{d}$ (Euclidean space), the dot product with the matrix-vector product satisfies $c\cdot(Gc)>0$. Call a real $d\times d$ matrix $T$ lower triangular if its entries satisfy $T_{il}=0$ whenever $l>i$.

1. (Existence and uniqueness) There is exactly one lower triangular $d\times d$ matrix $T$ with positive diagonal entries such that, with the matrix product and the identity matrix $I_d$,

2. (Invertibility) This $T$ is invertible, and its inverse is lower triangular with positive diagonal entries.

3. (Stability) Let $(G^{(k)})_{k\in\mathbb{N}}$ be a sequence of real symmetric $d\times d$ matrices that are positive semidefinite, meaning $c\cdot(G^{(k)}c)\ge0$ for every $c\in\mathbb{R}^{d}$, such that every entry $G^{(k)}_{il}$ has limit $G_{il}$ as $k\to\infty$. Then there is $K\in\mathbb{N}$ such that $G^{(k)}$ is positive definite for every $k\ge K$, and, for $k\ge K$, the matrices $T^{(k)}$ associated to $G^{(k)}$ by claim 1 satisfy $T^{(k)}_{il}\to T_{il}$ as $k\to\infty$ for all $i,l$.