recovering vectors from a Gram matrix

Question

I am working with a set of $n$ unit points on the unit sphere $S2$ and have 2 questions relating to their Gram matrix and certain rotations.(here $n=5$)

Let

$$ q_n = [[\tfrac{1}{3}, \tfrac{2}{3}, -\tfrac{2}{3}], [\tfrac{4}{9}, \tfrac{7}{9}, -\tfrac{4}{9}], [\tfrac{2}{11}, -\tfrac{6}{11}, \tfrac{9}{11}], [-\tfrac{2}{7}, \tfrac{3}{7}, \tfrac{6}{7}], [\tfrac{1}{9}, -\tfrac{4}{9}, \tfrac{8}{9}]]$$

be the first set of 5 unit points on the sphere.

Then

$$ G_q = Gram(q) = \begin{pmatrix} 1 & 26/27 & -28/33 & -8/21 & -23/27 \\ 26/27& 1 & -70/99 & -11/63 & -56/81 \\ -28/33 & -70/99 & 1 & 32/77 & 98/99 \\ -8/21 & -11/63 & 32/77 & 1 & 34/63 \\ -23/27 & -56/81 & 98/99 & 34/63 & 1 \end{pmatrix}$$

which is a symmetric matrix. We also note that if we let $V$ = $3\times5$ matrix from $q_n$, then $G_V = V^T V$ which is well known.

I used an algorithm to find an isotomy of the 5 unit points.

Let those points be

$$ s_n = [[1, 0, 0], [\tfrac{26}{27}, \tfrac{\sqrt{53}}{27}, 0], [-\tfrac{28}{33}, \tfrac{98\sqrt{53}}{1749}, \tfrac{27\sqrt{53}}{583}], [-\tfrac{8}{21}, \tfrac{109\sqrt{53}}{1113}, -\tfrac{30\sqrt{53}}{371}], [-\tfrac{23}{27}, \tfrac{94\sqrt{53}}{1431}, \tfrac{14\sqrt{53}}{477}]]$$

then we can check and see that

$\qquad G_n$ = Gram$(s_n)$ = Gram$(q_n)$

This is because any 2 isotomies have the same Gram matrix, i.e., we can rotate the unit points to any configuration on the unit sphere, keeping the angles and distances intact, but there are an infinite amount of ways to do this.

There has been a previous posting on the set of vectors and their Gram matrix, see Recovering a set of vectors from their Gram matrix but it doesn't answer my first question below.

My two questions:

1. Is there an algorithm which takes $G_{n=5}$ which is $5\times 5$ and returns an $3\times5$ isotomy matrix of $q_n$? Answers to the previous post said to use Choleksy factorization, but did not specify or explain how to specifically recover the points from $L^T$. Nor is it apparent to me how to use the Cholesky factorization which is a $n\times n$ matrix to recover the $3\times n$ point list.

Could you carefully show how to recover the $3\times n$ matrix from the $n\times n$ Gram matrix or $n\times n$ Cholesky factorization?

2. How can one determine the $R=SO(3)$ rotation which takes the first 3 points of s and returns the first 3 points of q? I am reasonably guessing that this particular $R=SO(3)$ will take $R(s) = q$, is that correct? I have determined that specifying 3 unique points on the unit sphere can correctly specify the needed rotation matrix $R=SO(3)$.

Although I have an operator which takes a single point $O(s_1)->q_1$, it does not work for all the points, i.e. $O(s_n)\neq q_n$.

My goal is twofold:

1. Using a Gram matrix G, recover a set of points as an isometry $p_n$.
2. Rotate the isometry $p_n$ to original configuration $q_n$ (is a reflection sometimes necessary, besides a rotation or two?)

I am attempting to take a set of unit points, find the Gram matrix, twiddle the Gram matrix entries slightly and restore another set of unit points (which will be close to the original set) after a rotation.

My goal is to converge the unit points to a specific arrangement and the Gram matrix shows groups of inner products which has identical values, hinting at the groups which can recreate the unit points which I am seeking as optimal arrangements.

Answer 1

$$ P^T H P = D $$ $$\left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ - \frac{ 26 }{ 27 } & 1 & 0 & 0 & 0 \\ \frac{ 1344 }{ 583 } & - \frac{ 882 }{ 583 } & 1 & 0 & 0 \\ \frac{ 146 }{ 21 } & - \frac{ 37 }{ 7 } & \frac{ 110 }{ 63 } & 1 & 0 \\ \frac{ 89 }{ 81 } & - \frac{ 22 }{ 27 } & - \frac{ 154 }{ 243 } & 0 & 1 \\ \end{array} \right) \left( \begin{array}{rrrrr} 6237 & 6006 & - 5292 & - 2376 & - 5313 \\ 6006 & 6237 & - 4410 & - 1089 & - 4312 \\ - 5292 & - 4410 & 6237 & 2592 & 6174 \\ - 2376 & - 1089 & 2592 & 6237 & 3366 \\ - 5313 & - 4312 & 6174 & 3366 & 6237 \\ \end{array} \right) \left( \begin{array}{rrrrr} 1 & - \frac{ 26 }{ 27 } & \frac{ 1344 }{ 583 } & \frac{ 146 }{ 21 } & \frac{ 89 }{ 81 } \\ 0 & 1 & - \frac{ 882 }{ 583 } & - \frac{ 37 }{ 7 } & - \frac{ 22 }{ 27 } \\ 0 & 0 & 1 & \frac{ 110 }{ 63 } & - \frac{ 154 }{ 243 } \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrrr} 6237 & 0 & 0 & 0 & 0 \\ 0 & \frac{ 4081 }{ 9 } & 0 & 0 & 0 \\ 0 & 0 & \frac{ 413343 }{ 583 } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ \frac{ 26 }{ 27 } & 1 & 0 & 0 & 0 \\ - \frac{ 28 }{ 33 } & \frac{ 882 }{ 583 } & 1 & 0 & 0 \\ - \frac{ 8 }{ 21 } & \frac{ 981 }{ 371 } & - \frac{ 110 }{ 63 } & 1 & 0 \\ - \frac{ 23 }{ 27 } & \frac{ 94 }{ 53 } & \frac{ 154 }{ 243 } & 0 & 1 \\ \end{array} \right) \left( \begin{array}{rrrrr} 6237 & 0 & 0 & 0 & 0 \\ 0 & \frac{ 4081 }{ 9 } & 0 & 0 & 0 \\ 0 & 0 & \frac{ 413343 }{ 583 } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right) \left( \begin{array}{rrrrr} 1 & \frac{ 26 }{ 27 } & - \frac{ 28 }{ 33 } & - \frac{ 8 }{ 21 } & - \frac{ 23 }{ 27 } \\ 0 & 1 & \frac{ 882 }{ 583 } & \frac{ 981 }{ 371 } & \frac{ 94 }{ 53 } \\ 0 & 0 & 1 & - \frac{ 110 }{ 63 } & \frac{ 154 }{ 243 } \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrrr} 6237 & 6006 & - 5292 & - 2376 & - 5313 \\ 6006 & 6237 & - 4410 & - 1089 & - 4312 \\ - 5292 & - 4410 & 6237 & 2592 & 6174 \\ - 2376 & - 1089 & 2592 & 6237 & 3366 \\ - 5313 & - 4312 & 6174 & 3366 & 6237 \\ \end{array} \right) $$

Algorithm discussed at http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr
https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia
$$ H = \left( \begin{array}{rrrrr} 6237 & 6006 & - 5292 & - 2376 & - 5313 \\ 6006 & 6237 & - 4410 & - 1089 & - 4312 \\ - 5292 & - 4410 & 6237 & 2592 & 6174 \\ - 2376 & - 1089 & 2592 & 6237 & 3366 \\ - 5313 & - 4312 & 6174 & 3366 & 6237 \\ \end{array} \right) $$ $$ D_0 = H $$ $$ E_j^T D_{j-1} E_j = D_j $$ $$ P_{j-1} E_j = P_j $$ $$ E_j^{-1} Q_{j-1} = Q_j $$ $$ P_j Q_j = Q_j P_j = I $$ $$ P_j^T H P_j = D_j $$ $$ Q_j^T D_j Q_j = H $$

$$ H = \left( \begin{array}{rrrrr} 6237 & 6006 & - 5292 & - 2376 & - 5313 \\ 6006 & 6237 & - 4410 & - 1089 & - 4312 \\ - 5292 & - 4410 & 6237 & 2592 & 6174 \\ - 2376 & - 1089 & 2592 & 6237 & 3366 \\ - 5313 & - 4312 & 6174 & 3366 & 6237 \\ \end{array} \right) $$

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$$ E_{1} = \left( \begin{array}{rrrrr} 1 & - \frac{ 26 }{ 27 } & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{1} = \left( \begin{array}{rrrrr} 1 & - \frac{ 26 }{ 27 } & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{1} = \left( \begin{array}{rrrrr} 1 & \frac{ 26 }{ 27 } & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{1} = \left( \begin{array}{rrrrr} 6237 & 0 & - 5292 & - 2376 & - 5313 \\ 0 & \frac{ 4081 }{ 9 } & 686 & 1199 & \frac{ 7238 }{ 9 } \\ - 5292 & 686 & 6237 & 2592 & 6174 \\ - 2376 & 1199 & 2592 & 6237 & 3366 \\ - 5313 & \frac{ 7238 }{ 9 } & 6174 & 3366 & 6237 \\ \end{array} \right) $$

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$$ E_{2} = \left( \begin{array}{rrrrr} 1 & 0 & \frac{ 28 }{ 33 } & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{2} = \left( \begin{array}{rrrrr} 1 & - \frac{ 26 }{ 27 } & \frac{ 28 }{ 33 } & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{2} = \left( \begin{array}{rrrrr} 1 & \frac{ 26 }{ 27 } & - \frac{ 28 }{ 33 } & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{2} = \left( \begin{array}{rrrrr} 6237 & 0 & 0 & - 2376 & - 5313 \\ 0 & \frac{ 4081 }{ 9 } & 686 & 1199 & \frac{ 7238 }{ 9 } \\ 0 & 686 & \frac{ 19215 }{ 11 } & 576 & 1666 \\ - 2376 & 1199 & 576 & 6237 & 3366 \\ - 5313 & \frac{ 7238 }{ 9 } & 1666 & 3366 & 6237 \\ \end{array} \right) $$

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$$ E_{3} = \left( \begin{array}{rrrrr} 1 & 0 & 0 & \frac{ 8 }{ 21 } & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{3} = \left( \begin{array}{rrrrr} 1 & - \frac{ 26 }{ 27 } & \frac{ 28 }{ 33 } & \frac{ 8 }{ 21 } & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{3} = \left( \begin{array}{rrrrr} 1 & \frac{ 26 }{ 27 } & - \frac{ 28 }{ 33 } & - \frac{ 8 }{ 21 } & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{3} = \left( \begin{array}{rrrrr} 6237 & 0 & 0 & 0 & - 5313 \\ 0 & \frac{ 4081 }{ 9 } & 686 & 1199 & \frac{ 7238 }{ 9 } \\ 0 & 686 & \frac{ 19215 }{ 11 } & 576 & 1666 \\ 0 & 1199 & 576 & \frac{ 37323 }{ 7 } & 1342 \\ - 5313 & \frac{ 7238 }{ 9 } & 1666 & 1342 & 6237 \\ \end{array} \right) $$

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$$ E_{4} = \left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & \frac{ 23 }{ 27 } \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{4} = \left( \begin{array}{rrrrr} 1 & - \frac{ 26 }{ 27 } & \frac{ 28 }{ 33 } & \frac{ 8 }{ 21 } & \frac{ 23 }{ 27 } \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{4} = \left( \begin{array}{rrrrr} 1 & \frac{ 26 }{ 27 } & - \frac{ 28 }{ 33 } & - \frac{ 8 }{ 21 } & - \frac{ 23 }{ 27 } \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{4} = \left( \begin{array}{rrrrr} 6237 & 0 & 0 & 0 & 0 \\ 0 & \frac{ 4081 }{ 9 } & 686 & 1199 & \frac{ 7238 }{ 9 } \\ 0 & 686 & \frac{ 19215 }{ 11 } & 576 & 1666 \\ 0 & 1199 & 576 & \frac{ 37323 }{ 7 } & 1342 \\ 0 & \frac{ 7238 }{ 9 } & 1666 & 1342 & \frac{ 15400 }{ 9 } \\ \end{array} \right) $$

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$$ E_{5} = \left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{ 882 }{ 583 } & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{5} = \left( \begin{array}{rrrrr} 1 & - \frac{ 26 }{ 27 } & \frac{ 1344 }{ 583 } & \frac{ 8 }{ 21 } & \frac{ 23 }{ 27 } \\ 0 & 1 & - \frac{ 882 }{ 583 } & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{5} = \left( \begin{array}{rrrrr} 1 & \frac{ 26 }{ 27 } & - \frac{ 28 }{ 33 } & - \frac{ 8 }{ 21 } & - \frac{ 23 }{ 27 } \\ 0 & 1 & \frac{ 882 }{ 583 } & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{5} = \left( \begin{array}{rrrrr} 6237 & 0 & 0 & 0 & 0 \\ 0 & \frac{ 4081 }{ 9 } & 0 & 1199 & \frac{ 7238 }{ 9 } \\ 0 & 0 & \frac{ 413343 }{ 583 } & - \frac{ 65610 }{ 53 } & \frac{ 23814 }{ 53 } \\ 0 & 1199 & - \frac{ 65610 }{ 53 } & \frac{ 37323 }{ 7 } & 1342 \\ 0 & \frac{ 7238 }{ 9 } & \frac{ 23814 }{ 53 } & 1342 & \frac{ 15400 }{ 9 } \\ \end{array} \right) $$

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$$ E_{6} = \left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & - \frac{ 981 }{ 371 } & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{6} = \left( \begin{array}{rrrrr} 1 & - \frac{ 26 }{ 27 } & \frac{ 1344 }{ 583 } & \frac{ 1086 }{ 371 } & \frac{ 23 }{ 27 } \\ 0 & 1 & - \frac{ 882 }{ 583 } & - \frac{ 981 }{ 371 } & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{6} = \left( \begin{array}{rrrrr} 1 & \frac{ 26 }{ 27 } & - \frac{ 28 }{ 33 } & - \frac{ 8 }{ 21 } & - \frac{ 23 }{ 27 } \\ 0 & 1 & \frac{ 882 }{ 583 } & \frac{ 981 }{ 371 } & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{6} = \left( \begin{array}{rrrrr} 6237 & 0 & 0 & 0 & 0 \\ 0 & \frac{ 4081 }{ 9 } & 0 & 0 & \frac{ 7238 }{ 9 } \\ 0 & 0 & \frac{ 413343 }{ 583 } & - \frac{ 65610 }{ 53 } & \frac{ 23814 }{ 53 } \\ 0 & 0 & - \frac{ 65610 }{ 53 } & \frac{ 801900 }{ 371 } & - \frac{ 41580 }{ 53 } \\ 0 & \frac{ 7238 }{ 9 } & \frac{ 23814 }{ 53 } & - \frac{ 41580 }{ 53 } & \frac{ 15400 }{ 9 } \\ \end{array} \right) $$

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$$ E_{7} = \left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & - \frac{ 94 }{ 53 } \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{7} = \left( \begin{array}{rrrrr} 1 & - \frac{ 26 }{ 27 } & \frac{ 1344 }{ 583 } & \frac{ 1086 }{ 371 } & \frac{ 407 }{ 159 } \\ 0 & 1 & - \frac{ 882 }{ 583 } & - \frac{ 981 }{ 371 } & - \frac{ 94 }{ 53 } \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{7} = \left( \begin{array}{rrrrr} 1 & \frac{ 26 }{ 27 } & - \frac{ 28 }{ 33 } & - \frac{ 8 }{ 21 } & - \frac{ 23 }{ 27 } \\ 0 & 1 & \frac{ 882 }{ 583 } & \frac{ 981 }{ 371 } & \frac{ 94 }{ 53 } \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{7} = \left( \begin{array}{rrrrr} 6237 & 0 & 0 & 0 & 0 \\ 0 & \frac{ 4081 }{ 9 } & 0 & 0 & 0 \\ 0 & 0 & \frac{ 413343 }{ 583 } & - \frac{ 65610 }{ 53 } & \frac{ 23814 }{ 53 } \\ 0 & 0 & - \frac{ 65610 }{ 53 } & \frac{ 801900 }{ 371 } & - \frac{ 41580 }{ 53 } \\ 0 & 0 & \frac{ 23814 }{ 53 } & - \frac{ 41580 }{ 53 } & \frac{ 15092 }{ 53 } \\ \end{array} \right) $$

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$$ E_{8} = \left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{ 110 }{ 63 } & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{8} = \left( \begin{array}{rrrrr} 1 & - \frac{ 26 }{ 27 } & \frac{ 1344 }{ 583 } & \frac{ 146 }{ 21 } & \frac{ 407 }{ 159 } \\ 0 & 1 & - \frac{ 882 }{ 583 } & - \frac{ 37 }{ 7 } & - \frac{ 94 }{ 53 } \\ 0 & 0 & 1 & \frac{ 110 }{ 63 } & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{8} = \left( \begin{array}{rrrrr} 1 & \frac{ 26 }{ 27 } & - \frac{ 28 }{ 33 } & - \frac{ 8 }{ 21 } & - \frac{ 23 }{ 27 } \\ 0 & 1 & \frac{ 882 }{ 583 } & \frac{ 981 }{ 371 } & \frac{ 94 }{ 53 } \\ 0 & 0 & 1 & - \frac{ 110 }{ 63 } & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{8} = \left( \begin{array}{rrrrr} 6237 & 0 & 0 & 0 & 0 \\ 0 & \frac{ 4081 }{ 9 } & 0 & 0 & 0 \\ 0 & 0 & \frac{ 413343 }{ 583 } & 0 & \frac{ 23814 }{ 53 } \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{ 23814 }{ 53 } & 0 & \frac{ 15092 }{ 53 } \\ \end{array} \right) $$

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$$ E_{9} = \left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & - \frac{ 154 }{ 243 } \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{9} = \left( \begin{array}{rrrrr} 1 & - \frac{ 26 }{ 27 } & \frac{ 1344 }{ 583 } & \frac{ 146 }{ 21 } & \frac{ 89 }{ 81 } \\ 0 & 1 & - \frac{ 882 }{ 583 } & - \frac{ 37 }{ 7 } & - \frac{ 22 }{ 27 } \\ 0 & 0 & 1 & \frac{ 110 }{ 63 } & - \frac{ 154 }{ 243 } \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{9} = \left( \begin{array}{rrrrr} 1 & \frac{ 26 }{ 27 } & - \frac{ 28 }{ 33 } & - \frac{ 8 }{ 21 } & - \frac{ 23 }{ 27 } \\ 0 & 1 & \frac{ 882 }{ 583 } & \frac{ 981 }{ 371 } & \frac{ 94 }{ 53 } \\ 0 & 0 & 1 & - \frac{ 110 }{ 63 } & \frac{ 154 }{ 243 } \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{9} = \left( \begin{array}{rrrrr} 6237 & 0 & 0 & 0 & 0 \\ 0 & \frac{ 4081 }{ 9 } & 0 & 0 & 0 \\ 0 & 0 & \frac{ 413343 }{ 583 } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right) $$

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$$ P^T H P = D $$ $$\left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ - \frac{ 26 }{ 27 } & 1 & 0 & 0 & 0 \\ \frac{ 1344 }{ 583 } & - \frac{ 882 }{ 583 } & 1 & 0 & 0 \\ \frac{ 146 }{ 21 } & - \frac{ 37 }{ 7 } & \frac{ 110 }{ 63 } & 1 & 0 \\ \frac{ 89 }{ 81 } & - \frac{ 22 }{ 27 } & - \frac{ 154 }{ 243 } & 0 & 1 \\ \end{array} \right) \left( \begin{array}{rrrrr} 6237 & 6006 & - 5292 & - 2376 & - 5313 \\ 6006 & 6237 & - 4410 & - 1089 & - 4312 \\ - 5292 & - 4410 & 6237 & 2592 & 6174 \\ - 2376 & - 1089 & 2592 & 6237 & 3366 \\ - 5313 & - 4312 & 6174 & 3366 & 6237 \\ \end{array} \right) \left( \begin{array}{rrrrr} 1 & - \frac{ 26 }{ 27 } & \frac{ 1344 }{ 583 } & \frac{ 146 }{ 21 } & \frac{ 89 }{ 81 } \\ 0 & 1 & - \frac{ 882 }{ 583 } & - \frac{ 37 }{ 7 } & - \frac{ 22 }{ 27 } \\ 0 & 0 & 1 & \frac{ 110 }{ 63 } & - \frac{ 154 }{ 243 } \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrrr} 6237 & 0 & 0 & 0 & 0 \\ 0 & \frac{ 4081 }{ 9 } & 0 & 0 & 0 \\ 0 & 0 & \frac{ 413343 }{ 583 } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ \frac{ 26 }{ 27 } & 1 & 0 & 0 & 0 \\ - \frac{ 28 }{ 33 } & \frac{ 882 }{ 583 } & 1 & 0 & 0 \\ - \frac{ 8 }{ 21 } & \frac{ 981 }{ 371 } & - \frac{ 110 }{ 63 } & 1 & 0 \\ - \frac{ 23 }{ 27 } & \frac{ 94 }{ 53 } & \frac{ 154 }{ 243 } & 0 & 1 \\ \end{array} \right) \left( \begin{array}{rrrrr} 6237 & 0 & 0 & 0 & 0 \\ 0 & \frac{ 4081 }{ 9 } & 0 & 0 & 0 \\ 0 & 0 & \frac{ 413343 }{ 583 } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right) \left( \begin{array}{rrrrr} 1 & \frac{ 26 }{ 27 } & - \frac{ 28 }{ 33 } & - \frac{ 8 }{ 21 } & - \frac{ 23 }{ 27 } \\ 0 & 1 & \frac{ 882 }{ 583 } & \frac{ 981 }{ 371 } & \frac{ 94 }{ 53 } \\ 0 & 0 & 1 & - \frac{ 110 }{ 63 } & \frac{ 154 }{ 243 } \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrrr} 6237 & 6006 & - 5292 & - 2376 & - 5313 \\ 6006 & 6237 & - 4410 & - 1089 & - 4312 \\ - 5292 & - 4410 & 6237 & 2592 & 6174 \\ - 2376 & - 1089 & 2592 & 6237 & 3366 \\ - 5313 & - 4312 & 6174 & 3366 & 6237 \\ \end{array} \right) $$

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