SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mathbf {X} = \left[ \begin{array}{l l l} 0 & 0 & a \\ 0 & 0 & b \\ a & b & 0 \end{array} \right], \tag {10}
$$
the singular value decomposition of  $\mathbf{X}$  is given as  $\mathbf{X} = \mathbf{U}\mathbf{S}\mathbf{W}^T$ , where
$$
\mathbf {U} = \left[ \begin{array}{c c c} \frac {a}{\sqrt {a ^ {2} + b ^ {2}}} & 0 & \frac {- b}{\sqrt {a ^ {2} + b ^ {2}}} \\ \frac {b}{\sqrt {a ^ {2} + b ^ {2}}} & 0 & \frac {a}{\sqrt {a ^ {2} + b ^ {2}}} \\ 0 & - 1 & 0 \end{array} \right], \mathbf {S} = \left[ \begin{array}{c c c} \sqrt {a ^ {2} + b ^ {2}} & 0 & 0 \\ 0 & \sqrt {a ^ {2} + b ^ {2}} & 0 \\ 0 & 0 & 0 \end{array} \right] a n d
$$
$$
\mathbf {W} = \left[ \begin{array}{c c c} 0 & \frac {- a}{\sqrt {a ^ {2} + b ^ {2}}} & \frac {- b}{\sqrt {a ^ {2} + b ^ {2}}} \\ 0 & \frac {- b}{\sqrt {a ^ {2} + b ^ {2}}} & \frac {a}{\sqrt {a ^ {2} + b ^ {2}}} \\ 1 & 0 & 0 \end{array} \right].
$$
Now, we prove the theorems stated in the main paper.
Theorem 1. For a set of consistent directions,  $\mathbf{v}_{ij}$ ,  $(i,j) \in \mathcal{E}_{\Delta}$ , in  $\mathcal{G}_{\Delta}$ , the absolute change in any eigenvalue of  $\mathbf{V}^{\mathbf{T}}\mathbf{V}$ , denoted as  $|\delta \lambda|$ , when the directions  $\mathbf{v}_{ij}$  are perturbed by small rotations  $\delta \mathbf{R}_{ij}$ , with  $\mathbf{n}_{ij}$  and  $\delta \theta_{ij} > 0$  being the rotation axis and angle, is bounded by
$$
| \delta \lambda | \leq \sum_ {(k, i, j) \in T I (\Delta)} \delta \theta_ {i j} \cdot \| \mathbf {v} _ {k i} ^ {\mathbf {n} _ {i j} \perp} \| \cdot \| \mathbf {v} _ {k j} ^ {\mathbf {n} _ {i j} \perp} \| \cdot \frac {\left| \sin \phi_ {(k , i) , (k , j)} ^ {\mathbf {n} _ {i j} \perp} \right|}{\sin^ {2} \phi_ {(k , i) , (k , j)}}.
$$
$$
\left[ \left(1 + (\mathbf {v} _ {i k} ^ {T} \mathbf {v} _ {j k}) ^ {2}\right) \left((\mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {i k}) ^ {2} + (\mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {j k}) ^ {2}\right) - 4 \cdot \mathbf {v} _ {i k} ^ {T} \mathbf {v} _ {j k} \cdot \mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {i k} \cdot \mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {j k} \right] ^ {\frac {1}{2}}, \qquad (2)
$$
Proof. Our goal is to check for the absolute change in the eigenvalue of  $\mathbf{V}^{\mathrm{T}}\mathbf{V}$ , denoted as  $|\delta \lambda|$ .  $\mathbf{V}^{\mathrm{T}}\mathbf{V}$  is a symmetric matrix. Using Result 1 and Eqn. 9, finding the matrix-2 norm of the perturbation on  $\mathbf{V}^{\mathrm{T}}\mathbf{V}$  will give us the bound on the change in its eigenvalue.
Let us first look at the case when one edge is perturbed and then extend it later to all edges. We note that  $\mathcal{E}_{\Delta} = \{(1,2),(2,3)(3,1)\}$ . Without loss of generality, we perturb  $\mathbf{v}_{31}$  and denote the perturbed vector as  $\mathbf{v}_{31}^{P}$ . Since the perturbation is made by a small rotation  $\delta \mathbf{R}_{31}$ , we can write the perturbed vector as
$$
\begin{array}{l} \mathbf {v} _ {3 1} ^ {P} = \delta \mathbf {R} _ {3 1} \mathbf {v} _ {3 1} \\ = \left(\mathbf {I} + \delta \theta_ {3 1} \left[ \mathbf {n} _ {3 1} \right] _ {\times}\right) \mathbf {v} _ {3 1} \\ = \mathbf {v} _ {3 1} + \delta \boldsymbol {\omega} _ {3 1} \times \mathbf {v} _ {3 1}, \tag {11} \\ \end{array}
$$
where  $\delta \omega_{31} = \delta \theta_{31}\mathbf{n}_{31}$ . From Eqn. 11, it can be seen that the perturbation can be treated as an additive perturbation  $\mathbf{v}_{31}^{P} = \mathbf{v}_{31} + \delta \mathbf{v}_{31}$ , where  $\delta \mathbf{v}_{31} = \delta \boldsymbol{\omega}_{31} \times \mathbf{v}_{31}$ . Let the perturbed matrix of the directions be  $\mathbf{V}^P$  such that  $\mathbf{V}^P = \mathbf{V} + \delta \mathbf{V}_{31}$ , where  $\delta \mathbf{V}_{31} = [0 \quad 0 \quad \delta \mathbf{v}_{31}]$ . Then,  $(\mathbf{V}^P)^T\mathbf{V}^P$  can be written as
$$
\begin{array}{l} \left(\mathbf {V} ^ {P}\right) ^ {T} \mathbf {V} ^ {P} = \left(\mathbf {V} + \delta \mathbf {V} _ {3 1}\right) ^ {T} \left(\mathbf {V} + \delta \mathbf {V} _ {3 1}\right) \\ \approx \mathbf {V} ^ {T} \mathbf {V} + \mathbf {V} ^ {T} \delta \mathbf {V} _ {3 1} + \delta \mathbf {V} _ {3 1} ^ {T} \mathbf {V} [ \text {i n g o r i n g s e c o n d o r d e r t e r m s} ] \\ = \mathbf {V} ^ {T} \mathbf {V} + \delta_ {3 1} \left(\mathbf {V} ^ {T} \mathbf {V}\right), \tag {12} \\ \end{array}
$$
where  $\delta_{31}(\mathbf{V}^T\mathbf{V}) = \mathbf{V}^T\delta \mathbf{V}_{31} + \delta \mathbf{V}_{31}^T\mathbf{V}$ . After simplification of  $\delta_{31}(\mathbf{V}^T\mathbf{V})$ , we get
$$
\delta_ {3 1} \left(\mathbf {V} ^ {T} \mathbf {V}\right) = \left[ \begin{array}{c c c} 0 & 0 & \delta \boldsymbol {\omega} _ {3 1} ^ {T} \left(\mathbf {v} _ {3 1} \times \mathbf {v} _ {1 2}\right) \\ 0 & 0 & \delta \boldsymbol {\omega} _ {3 1} ^ {T} \left(\mathbf {v} _ {3 1} \times \mathbf {v} _ {2 3}\right) \\ \delta \boldsymbol {\omega} _ {3 1} ^ {T} \left(\mathbf {v} _ {3 1} \times \mathbf {v} _ {1 2}\right) & \delta \boldsymbol {\omega} _ {3 1} ^ {T} \left(\mathbf {v} _ {3 1} \times \mathbf {v} _ {2 3}\right) & 0 \end{array} \right]. \tag {13}
$$
It can be seen that  $\delta_{31}(\mathbf{V}^T\mathbf{V})$  has the same structure as that of the matrix in Eqn. 10. So, using Result 2, we get
$$
\sigma_ {m a x} \left(\delta_ {3 1} \left(\mathbf {V} ^ {T} \mathbf {V}\right)\right) = \left\| \left[ \begin{array}{l} \delta \omega_ {3 1} ^ {T} \left(\mathbf {v} _ {3 1} \times \mathbf {v} _ {1 2}\right) \\ \delta \omega_ {3 1} ^ {T} \left(\mathbf {v} _ {3 1} \times \mathbf {v} _ {2 3}\right) \end{array} \right] \right\|. \tag {14}
$$
Since the directions are consistent, the three directions are coplanar. So, there exists  $\alpha$  and  $\beta$  such that
$$
\mathbf {v} _ {3 1} = \alpha \mathbf {v} _ {1 2} + \beta \mathbf {v} _ {2 3}. \tag {15}
$$
We obtain  $\alpha$  and  $\beta$  by premultiplying Eqn. 15 by  $\mathbf{v}_{12}^{T}$  and  $\mathbf{v}_{23}^{T}$  to get the following set of equations
$$
\mathbf {v} _ {1 2} ^ {T} \mathbf {v} _ {3 1} = \alpha \mathbf {v} _ {1 2} ^ {T} \mathbf {v} _ {1 2} + \beta \mathbf {v} _ {1 2} ^ {T} \mathbf {v} _ {2 3}, \tag {16}
$$
$$
\mathbf {v} _ {2 3} ^ {T} \mathbf {v} _ {3 1} = \alpha \mathbf {v} _ {2 3} ^ {T} \mathbf {v} _ {1 2} + \beta \mathbf {v} _ {2 3} ^ {T} \mathbf {v} _ {2 3}. \tag {17}
$$
Solving Eqns. 16 and 17 and using the fact that  $\mathbf{v}_{ij}^T\mathbf{v}_{ij} = 1$ , we get
$$
\alpha = \frac {\mathbf {v} _ {3 1} ^ {T} \mathbf {v} _ {1 2} - \left(\mathbf {v} _ {2 3} ^ {T} \mathbf {v} _ {3 1}\right) \left(\mathbf {v} _ {2 3} ^ {T} \mathbf {v} _ {1 2}\right)}{1 - \left(\mathbf {v} _ {1 2} ^ {T} \mathbf {v} _ {2 3}\right) ^ {2}}, \tag {18}
$$
$$
\beta = \frac {\mathbf {v} _ {3 1} ^ {T} \mathbf {v} _ {2 3} - \left(\mathbf {v} _ {1 2} ^ {T} \mathbf {v} _ {3 1}\right) \left(\mathbf {v} _ {1 2} ^ {T} \mathbf {v} _ {2 3}\right)}{1 - \left(\mathbf {v} _ {1 2} ^ {T} \mathbf {v} _ {2 3}\right) ^ {2}}. \tag {19}
$$
Also, using Eqn. 15, we get
$$
\mathbf {v} _ {3 1} \times \mathbf {v} _ {1 2} = \beta (\mathbf {v} _ {2 3} \times \mathbf {v} _ {1 2}), \tag {20}
$$
$$
\mathbf {v} _ {3 1} \times \mathbf {v} _ {2 3} = \alpha (\mathbf {v} _ {1 2} \times \mathbf {v} _ {2 3}). \tag {21}
$$
Using Eqns. 20 and 21 in Eqn. 14, we get
$$
\sigma_ {m a x} \left(\delta_ {3 1} \left(\mathbf {V} ^ {T} \mathbf {V}\right)\right) = \left\| \left[ \begin{array}{l} \beta \cdot \delta \omega_ {3 1} ^ {T} \left(\mathbf {v} _ {2 3} \times \mathbf {v} _ {1 2}\right) \\ \alpha \cdot \delta \omega_ {3 1} ^ {T} \left(\mathbf {v} _ {1 2} \times \mathbf {v} _ {2 3}\right) \end{array} \right] \right\| = \delta \theta_ {3 1} \left\| \left[ \begin{array}{l} \beta \cdot \mathbf {n} _ {3 1} ^ {T} \left(\mathbf {v} _ {2 3} \times \mathbf {v} _ {1 2}\right) \\ \alpha \cdot \mathbf {n} _ {3 1} ^ {T} \left(\mathbf {v} _ {1 2} \times \mathbf {v} _ {2 3}\right) \end{array} \right] \right\|. \tag {22}
$$
Let us decompose the directions as  $\mathbf{v} = \mathbf{v}^{\mathbf{n}\parallel} + \mathbf{v}^{\mathbf{n}\perp}$ , where  $\mathbf{v}^{\mathbf{n}\parallel}$  and  $\mathbf{v}^{\mathbf{n}\perp}$  denote parallel and perpendicular components of  $\mathbf{v}$  to  $\mathbf{n}$ , which can be obtained as
$$
\mathbf {v} ^ {\mathbf {n} \parallel} = \left(\mathbf {v} ^ {T} \mathbf {n}\right) \cdot \mathbf {n}, \tag {23}
$$
$$
\mathbf {v} ^ {\mathbf {n} \perp} = \mathbf {v} - \mathbf {v} ^ {\mathbf {n} \|} = \mathbf {v} - \left(\mathbf {v} ^ {T} \mathbf {n}\right) \cdot \mathbf {n}. \tag {24}
$$
Using Eqns. 23 and 24, we get
$$
\begin{array}{l} \mathbf {v} _ {2 3} \times \mathbf {v} _ {1 2} = \left(\left(\mathbf {v} _ {2 3} ^ {T} \mathbf {n} _ {3 1}\right) \cdot \mathbf {n} _ {3 1} + \mathbf {v} _ {2 3} ^ {\mathbf {n} _ {3 1} \perp}\right) \times \left(\left(\mathbf {v} _ {1 2} ^ {T} \mathbf {n} _ {3 1}\right) \cdot \mathbf {n} _ {3 1} + \mathbf {v} _ {1 2} ^ {\mathbf {n} _ {3 1} \perp}\right) \\ = \left(\left(\mathbf {v} _ {2 3} ^ {T} \mathbf {n} _ {3 1}\right) \cdot \mathbf {n} _ {3 1}\right) \times \left(\left(\mathbf {v} _ {1 2} ^ {T} \mathbf {n} _ {3 1}\right) \cdot \mathbf {n} _ {3 1}\right) + \left(\mathbf {v} _ {2 3} ^ {T} \mathbf {n} _ {3 1}\right) \cdot \mathbf {n} _ {3 1} \times \mathbf {v} _ {1 2} ^ {\mathbf {n} _ {3 1}} ^ {\perp} \\ + \mathbf {v} _ {2 3} ^ {\mathbf {n} _ {3 1} \perp} \times \left(\mathbf {v} _ {1 2} ^ {T} \mathbf {n} _ {3 1}\right) \cdot \mathbf {n} _ {3 1} + \mathbf {v} _ {2 3} ^ {\mathbf {n} _ {3 1} \perp} \times \mathbf {v} _ {1 2} ^ {\mathbf {n} _ {3 1} \perp} \\ = \left(\mathbf {v} _ {2 3} ^ {T} \mathbf {n} _ {3 1}\right) \cdot \left(\mathbf {n} _ {3 1} \times \mathbf {v} _ {1 2} ^ {\mathbf {n} _ {3 1} \perp}\right) + \left(\mathbf {v} _ {1 2} ^ {T} \mathbf {n} _ {3 1}\right) \cdot \left(\mathbf {v} _ {2 3} ^ {\mathbf {n} _ {3 1} \perp} \times \mathbf {n} _ {3 1}\right) + \left(\mathbf {v} _ {2 3} ^ {\mathbf {n} _ {3 1} \perp} \times \mathbf {v} _ {1 2} ^ {\mathbf {n} _ {3 1} \perp}\right). \tag {25} \\ \end{array}
$$
Premultiplying  $\mathbf{n}_{31}^T$  in Eqn. 25, we get
$$
\mathbf {n} _ {3 1} ^ {T} \left(\mathbf {v} _ {2 3} \times \mathbf {v} _ {1 2}\right) = \mathbf {n} _ {3 1} ^ {T} \left(\mathbf {v} _ {2 3} ^ {\mathbf {n} _ {3 1} \perp} \times \mathbf {v} _ {1 2} ^ {\mathbf {n} _ {3 1} \perp}\right) = \left\| \mathbf {v} _ {2 3} ^ {\mathbf {n} _ {3 1} \perp} \times \mathbf {v} _ {1 2} ^ {\mathbf {n} _ {3 1} \perp} \right\|, \tag {26}
$$
since, by construction,  $\mathbf{v}_{23}^{\mathbf{n}_{31}\perp}\times \mathbf{v}_{12}^{\mathbf{n}_{31}\perp} = \left\| \mathbf{v}_{23}^{\mathbf{n}_{31}\perp}\times \mathbf{v}_{12}^{\mathbf{n}_{31}\perp}\right\| \cdot \mathbf{n}_{31}$ . Using Eqn. 26 into Eqn. 22, we get
$$
\sigma_ {m a x} \left(\delta_ {3 1} \left(\mathbf {V} ^ {T} \mathbf {V}\right)\right) = \delta \theta_ {3 1} \cdot \left\| \mathbf {v} _ {1 2} ^ {\mathbf {n} _ {3 1} \perp} \times \mathbf {v} _ {2 3} ^ {\mathbf {n} _ {3 1} \perp} \right\| \cdot \left\| \left[ \begin{array}{c} - \beta \\ \alpha \end{array} \right] \right\|. \tag {27}
$$
Using the values of  $\alpha$  and  $\beta$  from Eqns. 18 and 19, we get
$$
\sigma_ {m a x} \left(\delta_ {3 1} \left(\mathbf {V} ^ {T} \mathbf {V}\right)\right) = \delta \theta_ {3 1} \cdot \frac {\left\| \mathbf {v} _ {1 2} ^ {\mathbf {n} _ {3 1} \perp} \times \mathbf {v} _ {2 3} ^ {\mathbf {n} _ {3 1} \perp} \right\|}{1 - \left(\mathbf {v} _ {1 2} ^ {T} \mathbf {v} _ {2 3}\right) ^ {2}}.
$$
$$
\begin{array}{l} \left[ \left(1 + \left(\mathbf {v} _ {1 2} ^ {T} \mathbf {v} _ {2 3}\right) ^ {2}\right) \cdot \left(\left(\mathbf {v} _ {2 3} ^ {T} \mathbf {v} _ {3 1}\right) ^ {2} + \left(\mathbf {v} _ {1 2} ^ {T} \mathbf {v} _ {3 1}\right) ^ {2}\right) - 4 \cdot \mathbf {v} _ {1 2} ^ {T} \mathbf {v} _ {2 3} \cdot \mathbf {v} _ {2 3} ^ {T} \mathbf {v} _ {3 1} \cdot \mathbf {v} _ {3 1} ^ {T} \mathbf {v} _ {1 2} \right] ^ {\frac {1}{2}} \\ = \delta \theta_ {3 1} \cdot \| \mathbf {v} _ {1 2} ^ {\mathbf {n} _ {3 1} \perp} \| \cdot \| \mathbf {v} _ {2 3} ^ {\mathbf {n} _ {3 1} \perp} \| \cdot \frac {| \sin \phi_ {(2 , 1) , (2 , 3)} ^ {\mathbf {n} _ {i j} \perp} |}{\sin \phi_ {(2 , 1) , (2 , 3)} ^ {2}}. \\ \end{array}
$$
$$
\left[ \left(1 + \left(\mathbf {v} _ {1 2} ^ {T} \mathbf {v} _ {2 3}\right) ^ {2}\right) \cdot \left(\left(\mathbf {v} _ {2 3} ^ {T} \mathbf {v} _ {3 1}\right) ^ {2} + \left(\mathbf {v} _ {1 2} ^ {T} \mathbf {v} _ {3 1}\right) ^ {2}\right) - 4 \cdot \mathbf {v} _ {1 2} ^ {T} \mathbf {v} _ {2 3} \cdot \mathbf {v} _ {2 3} ^ {T} \mathbf {v} _ {3 1} \cdot \mathbf {v} _ {3 1} ^ {T} \mathbf {v} _ {1 2} \right] ^ {\frac {1}{2}}, \tag {28}
$$
where  $\phi_{(2,1),(2,3)}^{\mathbf{n}_{31} \perp}$  is the angle between  $\mathbf{v}_{21}^{\mathbf{n}_{31} \perp}$  and  $\mathbf{v}_{23}^{\mathbf{n}_{31} \perp}$ , and  $\phi_{(2,1),(2,3)}$  is the angle between  $\mathbf{v}_{21}$  and  $\mathbf{v}_{23}$ .
Now, we perturb all three edges in  $\mathcal{G}_{\Delta}$ . Then, Eqn. 12 will change to
[ignoring second order terms]
$$
\begin{array}{l} \left(\mathbf {V} ^ {P}\right) ^ {T} \mathbf {V} ^ {P} = \left(\mathbf {V} + \delta \mathbf {V} _ {1 2} + \delta \mathbf {V} _ {2 3} + \delta \mathbf {V} _ {3 1}\right) ^ {T} \left(\mathbf {V} + \delta \mathbf {V} _ {1 2} + \delta \mathbf {V} _ {2 3} + \delta \mathbf {V} _ {3 1}\right) \\ \approx \mathbf {V} ^ {T} \mathbf {V} + \mathbf {V} ^ {T} \delta \mathbf {V} _ {1 2} + \delta \mathbf {V} _ {1 2} ^ {T} \mathbf {V} + \mathbf {V} ^ {T} \delta \mathbf {V} _ {2 3} + \delta \mathbf {V} _ {2 3} ^ {T} \mathbf {V} + \mathbf {V} ^ {T} \delta \mathbf {V} _ {3 1} + \delta \mathbf {V} _ {3 1} ^ {T} \mathbf {V} \\ = \mathbf {V} ^ {T} \mathbf {V} + \delta_ {1 2} \left(\mathbf {V} ^ {T} \mathbf {V}\right) + \delta_ {2 3} \left(\mathbf {V} ^ {T} \mathbf {V}\right) + \delta_ {3 1} \left(\mathbf {V} ^ {T} \mathbf {V}\right). \tag {29} \\ \end{array}
$$
Using Result 1, we get,
$$
\begin{array}{l} \left| \delta \lambda \right| \leq \left\| \delta_ {1 2} \left(\mathbf {V} ^ {T} \mathbf {V}\right) + \delta_ {2 3} \left(\mathbf {V} ^ {T} \mathbf {V}\right) + \delta_ {3 1} \left(\mathbf {V} ^ {T} \mathbf {V}\right) \right\| _ {2} \\ \leq \left\| \delta_ {1 2} \left(\mathbf {V} ^ {T} \mathbf {V}\right) \right\| _ {2} + \left\| \delta_ {2 3} \left(\mathbf {V} ^ {T} \mathbf {V}\right) \right\| _ {2} + \left\| \delta_ {3 1} \left(\mathbf {V} ^ {T} \mathbf {V}\right) \right\| _ {2} [ \text {u s i n g t r i a n g l e i n e q u a l i t y} ]. \tag {30} \\ \end{array}
$$
Following Eqn. 28 for each edge perturbation in Eqn. 30, we get
$$
\begin{array}{l} | \delta \lambda | \leq \sum_ {(k, i, j) \in T I (\Delta)} \delta \theta_ {i j} \cdot \| \mathbf {v} _ {k i} ^ {\mathbf {n} _ {i j} \perp} \| \cdot \| \mathbf {v} _ {k j} ^ {\mathbf {n} _ {i j} \perp} \| \cdot \frac {\left| \sin \phi_ {(k , i) , (k , j)} ^ {\mathbf {n} _ {i j} \perp} \right|}{\sin^ {2} \phi_ {(k , i) , (k , j)}}. \\ \left[ \left(1 + (\mathbf {v} _ {i k} ^ {T} \mathbf {v} _ {j k}) ^ {2}\right) \left((\mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {i k}) ^ {2} + (\mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {j k}) ^ {2}\right) - 4 \cdot \mathbf {v} _ {i k} ^ {T} \mathbf {v} _ {j k} \cdot \mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {i k} \cdot \mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {j k} \right] ^ {\frac {1}{2}}, \\ \end{array}
$$
where  $TI(\Delta) = \{(1,2,3),(2,3,1),(3,1,2)\}$ .
Corollary 1. For a set of consistent directions,  $\mathbf{v}_{ij}$ ,  $(i,j) \in \mathcal{E}_{\Delta}$ , in  $\mathcal{G}_{\Delta}$ , the absolute change in any eigenvalue of  $\mathbf{V}^{\mathbf{T}}\mathbf{V}$ , denoted as  $|\delta \lambda|$ , when the directions  $\mathbf{v}_{ij}$  are perturbed by small rotations  $\delta \mathbf{R}_{ij}$ , with  $\mathbf{n}_{ij}$  and  $\delta \theta_{ij} > 0$  being the rotation axis and angle, and  $\mathbf{n}_{ij}$  being orthogonal to  $\mathbf{v}_{ki}$  and  $\mathbf{v}_{kj}$  for all  $(i,j,k) \in TI(\Delta)$ , is bounded by
$$
\begin{array}{l} | \delta \lambda | \leq \sum_ {(k, i, j) \in T I (\Delta)} \delta \theta_ {i j} \cdot \frac {1}{\left| \sin \phi_ {(k , i) , (k , j)} \right|}. \\ \left[ \left(1 + \left(\mathbf {v} _ {i k} ^ {T} \mathbf {v} _ {j k}\right) ^ {2}\right) \left(\left(\mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {i k}\right) ^ {2} + \left(\mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {j k}\right) ^ {2}\right) - 4 \cdot \mathbf {v} _ {i k} ^ {T} \mathbf {v} _ {j k} \cdot \mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {i k} \cdot \mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {j k} \right] ^ {\frac {1}{2}}, \tag {3} \\ \end{array}
$$
Proof. The unperturbed directions  $\mathbf{v}_{ij}$  are coplanar since the directions are consistent. Given that the rotation axis  $\mathbf{n}_{ij}$  for all the rotations  $\delta \mathbf{R}_{ij}$  are orthogonal to  $\mathbf{v}_{ki}$  and  $\mathbf{v}_{kj}$  for all  $(i,j,k)\in TI(\Delta)$ , the rotation axis  $\mathbf{n}_{ij}$  is orthogonal to  $\mathbf{v}_{kl}$ , for all  $(i,j),(k,l)\in \mathcal{E}_{\Delta}$ . This ensures that the perturbed directions are still coplanar and, thus, are consistent. Since  $\mathbf{n}_{ij}$  is orthogonal to all the directions  $\mathbf{v}_{ij}$ , the component of directions parallel to the axis are zero, i.e.  $\mathbf{v}^{\mathrm{n}}\parallel = 0$ . So,  $\mathbf{v}^{\mathrm{n}\perp} = \mathbf{v}$  (refer to Eqns. 23 and 24) and  $\sin \phi_{(k,i),(k,j)}^{\mathbf{n}_{ij}\perp} = \sin \phi_{(k,i),(k,j)}$ . Also,  $\| \mathbf{v}_{ij}^{\mathrm{n}\perp}\| = \| \mathbf{v}_{ij}\| = 1$  for all  $(i,j)\in \mathcal{E}_{\Delta}$ . Hence, using these in Eqn. 2, we get
$$
\begin{array}{l} | \delta \lambda | \leq \sum_ {(k, i, j) \in T I (\Delta)} \delta \theta_ {i j} \cdot \frac {1}{\left| \sin \phi_ {(k , i) , (k , j)} \right|}. \\ \left[ \left(1 + (\mathbf {v} _ {i k} ^ {T} \mathbf {v} _ {j k}) ^ {2}\right) \left((\mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {i k}) ^ {2} + (\mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {j k}) ^ {2}\right) - 4 \cdot \mathbf {v} _ {i k} ^ {T} \mathbf {v} _ {j k} \cdot \mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {i k} \cdot \mathbf {v} _ {i j} ^ {T} \mathbf {v} _ {j k} \right] ^ {\frac {1}{2}}. \\ \end{array}
$$
Proof. The determinant of  $\mathbf{A}_{\mathcal{G}_{\Delta}}$  is given as  $|\mathbf{A}_{\mathcal{G}_{\Delta}}| = 2 \cdot \phi_{(1,2),(1,3)} \cdot \phi_{(2,1),(2,3)} \cdot \phi_{(3,1),(3,2)}$ . The determinant will be zero if and only if at least one of the angles in the triangle is zero. So, the matrix is non-singular when all the angles are non-zero.
It is known that the determinant is not a good measure of the closeness of the matrix to singularity [39]. So, we consider the closeness of any two columns in the matrix in terms of the cosine of the angles between the column vectors of  $\mathbf{A}_{\mathcal{G}_{\Delta}}$ . Let  $\mathbf{a}_{\mathcal{G}_{\Delta}}^{ij}$  be the column of  $\mathbf{A}_{\mathcal{G}_{\Delta}}$  corresponding to the edge  $(i,j)\in \mathcal{E}_{\Delta}$ . In  $\mathcal{G}_{\Delta}$ , there is one node common for any pair of edges. So, the cosine of the angle between any two columns is given as  $\cos \angle (\mathbf{a}_{\mathcal{G}_{\Delta}}^{ij},\mathbf{a}_{\mathcal{G}_{\Delta}}^{jk}) = \frac{\phi_{(i,j),(i,k)}\cdot\phi_{(k,i),(k,j)}}{\sqrt{\phi_{(i,j),(i,k)}^2 + \phi_{(j,i),(j,k)}^2}\cdot\sqrt{\phi_{(j,i),(j,k)}^2 + \phi_{(k,i),(k,j)}^2}}$ .
It can be clearly seen that if the angle at the common node  $j$  is small, i.e.  $\phi_{(j,i),(j,k)}\to 0$  radian, then the two columns are very close or  $\cos \angle (\mathbf{a}_{\mathcal{G}_{\Delta}}^{ij},\mathbf{a}_{\mathcal{G}_{\Delta}}^{jk})\to 1$ . Small values of  $\phi_{(j,i),(j,k)}$  make  $\mathbf{A}_{\mathcal{G}_{\Delta}}$  nearly singular and its condition number high. If all values of  $\phi_{(j,i),(j,k)}$  are sufficiently large, then  $\cos \angle (\mathbf{a}_{\mathcal{G}_{\Delta}}^{ij},\mathbf{a}_{\mathcal{G}_{\Delta}}^{jk})$  between any two columns will not be close to 1 making  $\mathbf{A}_{\mathcal{G}_{\Delta}}$  well conditioned with a low condition number. Thus, the condition number of  $\mathbf{A}_{\mathcal{G}_{\Delta}}$  is reflective of the skewness of the triangle formed from the directions in  $\mathcal{E}_{\Delta}$ .
Proof. Let us assume that all the angles in all the triangles formed by the triplets in  $\mathcal{G}$  are sufficiently large. In  $\mathbf{A}_{\mathcal{G}}$ , no two columns can have exactly the same set of non-zero elements. This is evident since the diagonal entries are zero. In real-world scenarios,  $\mathcal{G}$  is a sparse network, and thus  $\mathbf{A}_{\mathcal{G}}$  is sparse. If two columns have exactly the same set of non-zero elements, then it means the two edges share exactly the same set of triplets. This is attained only when the two edges are connecting the same two nodes. But this is not possible since multiple edges between nodes are not allowed. Also, every column will have at least two non-zero elements since every edge is a part of at least one triplet and angles measured with the common nodes between edges will be present. This makes the angle matrix non-singular when all the angles in all the triangles are sufficiently large.
Next, we look at the conditioning of the matrix. Let  $\mathbf{a}_{\mathcal{G}}^{ij}$  be the column of  $\mathbf{A}_{\mathcal{G}}$  corresponding to the edge  $(i,j)\in \mathcal{E}$ . We check the cosine of the angle between any two columns of  $\mathbf{A}_{\mathcal{G}}$  to understand the conditioning of  $\mathbf{A}_{\mathcal{G}}$ . We define triplets to be adjacent if they share a common edge, otherwise they are non-adjacent. There are only three cases which arise between any two edges  $(i,j)$  and  $(k,l)$ ,  $(i,j)$ ,  $(k,l)\in \mathcal{E}$ , and are listed below:
1.  $(i,j)$  and  $(k,l)$  belong to the same triplet,
2.  $(i,j)$  and  $(k,l)$  belong to adjacent triplets,
3.  $(i,j)$  and  $(k,l)$  belong to non-adjacent triplets.
Since each column in  $\mathbf{A}_{\mathcal{G}}$  represents an edge, there are only the three cases, stated above, which occur for the pair of columns. For all three cases, the cosine of angles between two columns is checked. Let  $Trp$  be the set of all triplets in  $\mathcal{G}$ .
Case 1: Edges  $(i,j)$  and  $(k,l)$  belong to the same triplet.
$$
\begin{array}{l} \cos \angle (\mathbf {a} _ {\mathcal {G}} ^ {i j}, \mathbf {a} _ {\mathcal {G}} ^ {k l}) \\ = \frac {\phi_ {(c _ {1} ^ {\prime} , c) , (c _ {1} ^ {\prime} , c _ {2} ^ {\prime})} \cdot \phi_ {(c _ {2} ^ {\prime} , c _ {1} ^ {\prime}) , (c _ {2} ^ {\prime} , c)}}{\sqrt {\sum_ {p : (i , j , p) \in T r p} \left(\phi_ {(i , j) , (i , p)} ^ {2} + \phi_ {(j , i) , (j , p)} ^ {2}\right)} \cdot \sqrt {\sum_ {q : (k , l , q) \in T r p} \left(\phi_ {(k , l) , (k , p)} ^ {2} + \phi_ {(l , k) , (l , q)} ^ {2}\right)}}, \\ \end{array}
$$
where  $c \in \{i, j\} \cap \{k, l\}$  is the common node and  $c_1', c_2' \in \{i, j\} \cup \{k, l\} \setminus \{c\}$  with  $c_1' \neq c_2'$  are the other two nodes in the triplet.
It can be seen that for this case, if all the angles are sufficiently large, then  $\cos \angle (\mathbf{a}_{\mathcal{G}}^{ij},\mathbf{a}_{\mathcal{G}}^{kl})$  cannot be close to 1, and thus the two columns are not similar.
Case 2: Edges  $(i,j)$  and  $(k,l)$  belong to adjacent triplets  $tr1, tr2 \in Trp$ .
Part I: When the edges  $(i,j)$  and  $(k,l)$  share a common node of the triplets  $tr1$  and  $tr2$ :
$$
\begin{array}{l} \cos \angle (\mathbf {a} _ {\mathcal {G}} ^ {i j}, \mathbf {a} _ {\mathcal {G}} ^ {k l}) \\ = \frac {\phi_ {(r , c) , (r , c _ {1} ^ {\prime})} \cdot \phi_ {(r , c) , (r , c _ {2} ^ {\prime})}}{\sqrt {\sum_ {p : (i , j , p) \in T r p} \left(\phi_ {(i , j) , (i , p)} ^ {2} + \phi_ {(j , i) , (j , p)} ^ {2}\right)} \cdot \sqrt {\sum_ {q : (k , l , q) \in T r p} \left(\phi_ {(k , l) , (k , p)} ^ {2} + \phi_ {(l , k) , (l , q)} ^ {2}\right)}}, \\ \end{array}
$$
where  $c \in \{i, j\} \cap \{k, l\}$  is the common node,  $(r, c) \in \mathcal{E}$  is the common edge between  $tr1$  and  $tr2$ , and  $c_1' \in \{i, j\} \setminus \{c\}$ ,  $c_2' \in \{k, l\} \setminus \{c\}$  are the other two nodes in the two triplets.
Part II: When the edges  $(i,j)$  and  $(k,l)$  are connected by the common edge of the triplets  $tr1$  and  $tr2$ :