TITLE: Checking positive semi-definiteness of integer matrix
QUESTION [7 upvotes]: Key Problem : Is there any theorem about eigenvalues or positive semi-definiteness of small size matrices with small integer elements?
  

I have to check positive semi-definiteness of many symmetric matrices with integer elements. First I used eigenvalues, but floating point round error happens : eig(in numpy) sometimes gives small negative eigenvalues when a matrix is actually positive semi-definite. 
I know Sylvester's criterion  for positive semi-definiteness can avoid this problem. But I really don't want to use it since it requires computation of determinant of all principal minors, and I have to deal with really many matrices($ > 10^{20}$). I have to do anything to reduce number of calculations.

All the elements are integer and have small absolute values($\ <3 $), sizes of matrices are also small($\ < 10 \times 10 $). It seems pretty good condition, so I believe someone already researched this kind of matrices. Does anyone know useful theorems for this situation?
P.S: I'm pretty newbie in English Internet community and not native English speaker. So if you find something awkward, pardon me and let me know.

REPLY [5 votes]: For small symmetric matrices, you could look at the characteristic polynomial.
The real symmetric matrix $A$ is positive semidefinite iff the coefficients of the characteristic polynomial are alternating in sign.  For $n \times n$ matrices this gives you $n$ integer expressions to check.

REPLY [2 votes]: I wrote a program that takes a square integer matrix $H$ and produces square rational $P$ such that $P^T H P = D$ is diagonal and rational. In case it matters, $\det P = \pm 1.$ The program outputs in Latex. By Sylvester's Law of Inertia, $H$ is positive definite if and only if $D$ is positive definite, and there is no approximation involved. It is really just repeated completing the square made up into a reverse direction algorithm by parties unknown. The algorithm is given in detail at http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr
The way I like to write this, we introduce one new "elementary" matrix at a time and keep track of various things. In the most fortunate outcome, only one type of matrix is used and $P$ is upper triangular, but this does not always happen.
Oh, this works fine with semi-definite matrices. If the diagonal $D$ has some positive entries and some (diagonal) zero entries, then $H$ is positive semi-definite. No guesswork.
Here is the input
~jagy@phobeusjunior:~$ ./matrix_congruence 5
input row number   1  here 0 1 2 3 4
0 1 2 3 4
input row number   2  here 1 5 6 7 8
1 5 6 7 8
input row number   3  here 2 6 9 10 11
2 6 9 10 11
input row number   4  here 3 7 10 12 13
3 7 10 12 13
input row number   5  here 4 8 11 13 14
4 8 11 13 14

$$ P^T H P = D  $$
$$\left( 
\begin{array}{rrrrr} 
0 & 1 & 0 & 0 & 0 \\ 
1 &  -  \frac{ 1 }{ 5 }  & 0 & 0 & 0 \\ 
4 &  - 2 & 1 & 0 & 0 \\ 
 \frac{ 8 }{ 5 }  &  \frac{ 1 }{ 5 }  &  -  \frac{ 8 }{ 5 }  & 1 & 0 \\ 
 \frac{ 8 }{ 11 }  &  \frac{ 1 }{ 11 }  &  \frac{ 3 }{ 11 }  &  -  \frac{ 17 }{ 11 }  & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrrr} 
0 & 1 & 2 & 3 & 4 \\ 
1 & 5 & 6 & 7 & 8 \\ 
2 & 6 & 9 & 10 & 11 \\ 
3 & 7 & 10 & 12 & 13 \\ 
4 & 8 & 11 & 13 & 14 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrrr} 
0 & 1 & 4 &  \frac{ 8 }{ 5 }  &  \frac{ 8 }{ 11 }  \\ 
1 &  -  \frac{ 1 }{ 5 }  &  - 2 &  \frac{ 1 }{ 5 }  &  \frac{ 1 }{ 11 }  \\ 
0 & 0 & 1 &  -  \frac{ 8 }{ 5 }  &  \frac{ 3 }{ 11 }  \\ 
0 & 0 & 0 & 1 &  -  \frac{ 17 }{ 11 }  \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rrrrr} 
5 & 0 & 0 & 0 & 0 \\ 
0 &  -  \frac{ 1 }{ 5 }  & 0 & 0 & 0 \\ 
0 & 0 & 5 & 0 & 0 \\ 
0 & 0 & 0 &  \frac{ 11 }{ 5 }  & 0 \\ 
0 & 0 & 0 & 0 &  \frac{ 6 }{ 11 }  \\ 
\end{array}
\right) 
$$
$$ D_0 = H  $$
$$ E_j^T D_{j-1} E_j = D_j  $$
$$  P_{j-1} E_j = P_j  $$
$$ E_j^{-1} Q_{j-1}  = Q_j  $$
$$  P_j Q_j = I  $$
$$ P_j^T H P_j = D_j  $$
$$ Q_j^T D_j Q_j = H  $$
$$ H = \left( 
\begin{array}{rrrrr} 
0 & 1 & 2 & 3 & 4 \\ 
1 & 5 & 6 & 7 & 8 \\ 
2 & 6 & 9 & 10 & 11 \\ 
3 & 7 & 10 & 12 & 13 \\ 
4 & 8 & 11 & 13 & 14 \\ 
\end{array}
\right) 
$$
==============================================
$$ E_{1} = \left( 
\begin{array}{rrrrr} 
0 & 1 & 0 & 0 & 0 \\ 
1 & 0 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
$$
$$  P_{1} = \left( 
\begin{array}{rrrrr} 
0 & 1 & 0 & 0 & 0 \\ 
1 & 0 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{1} = \left( 
\begin{array}{rrrrr} 
0 & 1 & 0 & 0 & 0 \\ 
1 & 0 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; D_{1} = \left( 
\begin{array}{rrrrr} 
5 & 1 & 6 & 7 & 8 \\ 
1 & 0 & 2 & 3 & 4 \\ 
6 & 2 & 9 & 10 & 11 \\ 
7 & 3 & 10 & 12 & 13 \\ 
8 & 4 & 11 & 13 & 14 \\ 
\end{array}
\right) 
$$
==============================================
$$ E_{2} = \left( 
\begin{array}{rrrrr} 
1 &  -  \frac{ 1 }{ 5 }  & 0 & 0 & 0 \\ 
0 & 1 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
$$
$$  P_{2} = \left( 
\begin{array}{rrrrr} 
0 & 1 & 0 & 0 & 0 \\ 
1 &  -  \frac{ 1 }{ 5 }  & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{2} = \left( 
\begin{array}{rrrrr} 
 \frac{ 1 }{ 5 }  & 1 & 0 & 0 & 0 \\ 
1 & 0 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; D_{2} = \left( 
\begin{array}{rrrrr} 
5 & 0 & 6 & 7 & 8 \\ 
0 &  -  \frac{ 1 }{ 5 }  &  \frac{ 4 }{ 5 }  &  \frac{ 8 }{ 5 }  &  \frac{ 12 }{ 5 }  \\ 
6 &  \frac{ 4 }{ 5 }  & 9 & 10 & 11 \\ 
7 &  \frac{ 8 }{ 5 }  & 10 & 12 & 13 \\ 
8 &  \frac{ 12 }{ 5 }  & 11 & 13 & 14 \\ 
\end{array}
\right) 
$$
==============================================
$$ E_{3} = \left( 
\begin{array}{rrrrr} 
1 & 0 &  -  \frac{ 6 }{ 5 }  & 0 & 0 \\ 
0 & 1 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
$$
$$  P_{3} = \left( 
\begin{array}{rrrrr} 
0 & 1 & 0 & 0 & 0 \\ 
1 &  -  \frac{ 1 }{ 5 }  &  -  \frac{ 6 }{ 5 }  & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{3} = \left( 
\begin{array}{rrrrr} 
 \frac{ 1 }{ 5 }  & 1 &  \frac{ 6 }{ 5 }  & 0 & 0 \\ 
1 & 0 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; D_{3} = \left( 
\begin{array}{rrrrr} 
5 & 0 & 0 & 7 & 8 \\ 
0 &  -  \frac{ 1 }{ 5 }  &  \frac{ 4 }{ 5 }  &  \frac{ 8 }{ 5 }  &  \frac{ 12 }{ 5 }  \\ 
0 &  \frac{ 4 }{ 5 }  &  \frac{ 9 }{ 5 }  &  \frac{ 8 }{ 5 }  &  \frac{ 7 }{ 5 }  \\ 
7 &  \frac{ 8 }{ 5 }  &  \frac{ 8 }{ 5 }  & 12 & 13 \\ 
8 &  \frac{ 12 }{ 5 }  &  \frac{ 7 }{ 5 }  & 13 & 14 \\ 
\end{array}
\right) 
$$
==============================================
$$ E_{4} = \left( 
\begin{array}{rrrrr} 
1 & 0 & 0 &  -  \frac{ 7 }{ 5 }  & 0 \\ 
0 & 1 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
$$
$$  P_{4} = \left( 
\begin{array}{rrrrr} 
0 & 1 & 0 & 0 & 0 \\ 
1 &  -  \frac{ 1 }{ 5 }  &  -  \frac{ 6 }{ 5 }  &  -  \frac{ 7 }{ 5 }  & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{4} = \left( 
\begin{array}{rrrrr} 
 \frac{ 1 }{ 5 }  & 1 &  \frac{ 6 }{ 5 }  &  \frac{ 7 }{ 5 }  & 0 \\ 
1 & 0 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; D_{4} = \left( 
\begin{array}{rrrrr} 
5 & 0 & 0 & 0 & 8 \\ 
0 &  -  \frac{ 1 }{ 5 }  &  \frac{ 4 }{ 5 }  &  \frac{ 8 }{ 5 }  &  \frac{ 12 }{ 5 }  \\ 
0 &  \frac{ 4 }{ 5 }  &  \frac{ 9 }{ 5 }  &  \frac{ 8 }{ 5 }  &  \frac{ 7 }{ 5 }  \\ 
0 &  \frac{ 8 }{ 5 }  &  \frac{ 8 }{ 5 }  &  \frac{ 11 }{ 5 }  &  \frac{ 9 }{ 5 }  \\ 
8 &  \frac{ 12 }{ 5 }  &  \frac{ 7 }{ 5 }  &  \frac{ 9 }{ 5 }  & 14 \\ 
\end{array}
\right) 
$$
==============================================
$$ E_{5} = \left( 
\begin{array}{rrrrr} 
1 & 0 & 0 & 0 &  -  \frac{ 8 }{ 5 }  \\ 
0 & 1 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
$$
$$  P_{5} = \left( 
\begin{array}{rrrrr} 
0 & 1 & 0 & 0 & 0 \\ 
1 &  -  \frac{ 1 }{ 5 }  &  -  \frac{ 6 }{ 5 }  &  -  \frac{ 7 }{ 5 }  &  -  \frac{ 8 }{ 5 }  \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{5} = \left( 
\begin{array}{rrrrr} 
 \frac{ 1 }{ 5 }  & 1 &  \frac{ 6 }{ 5 }  &  \frac{ 7 }{ 5 }  &  \frac{ 8 }{ 5 }  \\ 
1 & 0 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; D_{5} = \left( 
\begin{array}{rrrrr} 
5 & 0 & 0 & 0 & 0 \\ 
0 &  -  \frac{ 1 }{ 5 }  &  \frac{ 4 }{ 5 }  &  \frac{ 8 }{ 5 }  &  \frac{ 12 }{ 5 }  \\ 
0 &  \frac{ 4 }{ 5 }  &  \frac{ 9 }{ 5 }  &  \frac{ 8 }{ 5 }  &  \frac{ 7 }{ 5 }  \\ 
0 &  \frac{ 8 }{ 5 }  &  \frac{ 8 }{ 5 }  &  \frac{ 11 }{ 5 }  &  \frac{ 9 }{ 5 }  \\ 
0 &  \frac{ 12 }{ 5 }  &  \frac{ 7 }{ 5 }  &  \frac{ 9 }{ 5 }  &  \frac{ 6 }{ 5 }  \\ 
\end{array}
\right) 
$$
==============================================
$$ E_{6} = \left( 
\begin{array}{rrrrr} 
1 & 0 & 0 & 0 & 0 \\ 
0 & 1 & 4 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
$$
$$  P_{6} = \left( 
\begin{array}{rrrrr} 
0 & 1 & 4 & 0 & 0 \\ 
1 &  -  \frac{ 1 }{ 5 }  &  - 2 &  -  \frac{ 7 }{ 5 }  &  -  \frac{ 8 }{ 5 }  \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{6} = \left( 
\begin{array}{rrrrr} 
 \frac{ 1 }{ 5 }  & 1 &  \frac{ 6 }{ 5 }  &  \frac{ 7 }{ 5 }  &  \frac{ 8 }{ 5 }  \\ 
1 & 0 &  - 4 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; D_{6} = \left( 
\begin{array}{rrrrr} 
5 & 0 & 0 & 0 & 0 \\ 
0 &  -  \frac{ 1 }{ 5 }  & 0 &  \frac{ 8 }{ 5 }  &  \frac{ 12 }{ 5 }  \\ 
0 & 0 & 5 & 8 & 11 \\ 
0 &  \frac{ 8 }{ 5 }  & 8 &  \frac{ 11 }{ 5 }  &  \frac{ 9 }{ 5 }  \\ 
0 &  \frac{ 12 }{ 5 }  & 11 &  \frac{ 9 }{ 5 }  &  \frac{ 6 }{ 5 }  \\ 
\end{array}
\right) 
$$
==============================================
$$ E_{7} = \left( 
\begin{array}{rrrrr} 
1 & 0 & 0 & 0 & 0 \\ 
0 & 1 & 0 & 8 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
$$
$$  P_{7} = \left( 
\begin{array}{rrrrr} 
0 & 1 & 4 & 8 & 0 \\ 
1 &  -  \frac{ 1 }{ 5 }  &  - 2 &  - 3 &  -  \frac{ 8 }{ 5 }  \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{7} = \left( 
\begin{array}{rrrrr} 
 \frac{ 1 }{ 5 }  & 1 &  \frac{ 6 }{ 5 }  &  \frac{ 7 }{ 5 }  &  \frac{ 8 }{ 5 }  \\ 
1 & 0 &  - 4 &  - 8 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; D_{7} = \left( 
\begin{array}{rrrrr} 
5 & 0 & 0 & 0 & 0 \\ 
0 &  -  \frac{ 1 }{ 5 }  & 0 & 0 &  \frac{ 12 }{ 5 }  \\ 
0 & 0 & 5 & 8 & 11 \\ 
0 & 0 & 8 & 15 & 21 \\ 
0 &  \frac{ 12 }{ 5 }  & 11 & 21 &  \frac{ 6 }{ 5 }  \\ 
\end{array}
\right) 
$$
==============================================
$$ E_{8} = \left( 
\begin{array}{rrrrr} 
1 & 0 & 0 & 0 & 0 \\ 
0 & 1 & 0 & 0 & 12 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
$$
$$  P_{8} = \left( 
\begin{array}{rrrrr} 
0 & 1 & 4 & 8 & 12 \\ 
1 &  -  \frac{ 1 }{ 5 }  &  - 2 &  - 3 &  - 4 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{8} = \left( 
\begin{array}{rrrrr} 
 \frac{ 1 }{ 5 }  & 1 &  \frac{ 6 }{ 5 }  &  \frac{ 7 }{ 5 }  &  \frac{ 8 }{ 5 }  \\ 
1 & 0 &  - 4 &  - 8 &  - 12 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; D_{8} = \left( 
\begin{array}{rrrrr} 
5 & 0 & 0 & 0 & 0 \\ 
0 &  -  \frac{ 1 }{ 5 }  & 0 & 0 & 0 \\ 
0 & 0 & 5 & 8 & 11 \\ 
0 & 0 & 8 & 15 & 21 \\ 
0 & 0 & 11 & 21 & 30 \\ 
\end{array}
\right) 
$$
==============================================
$$ E_{9} = \left( 
\begin{array}{rrrrr} 
1 & 0 & 0 & 0 & 0 \\ 
0 & 1 & 0 & 0 & 0 \\ 
0 & 0 & 1 &  -  \frac{ 8 }{ 5 }  & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
$$
$$  P_{9} = \left( 
\begin{array}{rrrrr} 
0 & 1 & 4 &  \frac{ 8 }{ 5 }  & 12 \\ 
1 &  -  \frac{ 1 }{ 5 }  &  - 2 &  \frac{ 1 }{ 5 }  &  - 4 \\ 
0 & 0 & 1 &  -  \frac{ 8 }{ 5 }  & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{9} = \left( 
\begin{array}{rrrrr} 
 \frac{ 1 }{ 5 }  & 1 &  \frac{ 6 }{ 5 }  &  \frac{ 7 }{ 5 }  &  \frac{ 8 }{ 5 }  \\ 
1 & 0 &  - 4 &  - 8 &  - 12 \\ 
0 & 0 & 1 &  \frac{ 8 }{ 5 }  & 0 \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; D_{9} = \left( 
\begin{array}{rrrrr} 
5 & 0 & 0 & 0 & 0 \\ 
0 &  -  \frac{ 1 }{ 5 }  & 0 & 0 & 0 \\ 
0 & 0 & 5 & 0 & 11 \\ 
0 & 0 & 0 &  \frac{ 11 }{ 5 }  &  \frac{ 17 }{ 5 }  \\ 
0 & 0 & 11 &  \frac{ 17 }{ 5 }  & 30 \\ 
\end{array}
\right) 
$$
==============================================
$$ E_{10} = \left( 
\begin{array}{rrrrr} 
1 & 0 & 0 & 0 & 0 \\ 
0 & 1 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 &  -  \frac{ 11 }{ 5 }  \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
$$
$$  P_{10} = \left( 
\begin{array}{rrrrr} 
0 & 1 & 4 &  \frac{ 8 }{ 5 }  &  \frac{ 16 }{ 5 }  \\ 
1 &  -  \frac{ 1 }{ 5 }  &  - 2 &  \frac{ 1 }{ 5 }  &  \frac{ 2 }{ 5 }  \\ 
0 & 0 & 1 &  -  \frac{ 8 }{ 5 }  &  -  \frac{ 11 }{ 5 }  \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{10} = \left( 
\begin{array}{rrrrr} 
 \frac{ 1 }{ 5 }  & 1 &  \frac{ 6 }{ 5 }  &  \frac{ 7 }{ 5 }  &  \frac{ 8 }{ 5 }  \\ 
1 & 0 &  - 4 &  - 8 &  - 12 \\ 
0 & 0 & 1 &  \frac{ 8 }{ 5 }  &  \frac{ 11 }{ 5 }  \\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; D_{10} = \left( 
\begin{array}{rrrrr} 
5 & 0 & 0 & 0 & 0 \\ 
0 &  -  \frac{ 1 }{ 5 }  & 0 & 0 & 0 \\ 
0 & 0 & 5 & 0 & 0 \\ 
0 & 0 & 0 &  \frac{ 11 }{ 5 }  &  \frac{ 17 }{ 5 }  \\ 
0 & 0 & 0 &  \frac{ 17 }{ 5 }  &  \frac{ 29 }{ 5 }  \\ 
\end{array}
\right) 
$$
==============================================
$$ E_{11} = \left( 
\begin{array}{rrrrr} 
1 & 0 & 0 & 0 & 0 \\ 
0 & 1 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 &  -  \frac{ 17 }{ 11 }  \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
$$
$$  P_{11} = \left( 
\begin{array}{rrrrr} 
0 & 1 & 4 &  \frac{ 8 }{ 5 }  &  \frac{ 8 }{ 11 }  \\ 
1 &  -  \frac{ 1 }{ 5 }  &  - 2 &  \frac{ 1 }{ 5 }  &  \frac{ 1 }{ 11 }  \\ 
0 & 0 & 1 &  -  \frac{ 8 }{ 5 }  &  \frac{ 3 }{ 11 }  \\ 
0 & 0 & 0 & 1 &  -  \frac{ 17 }{ 11 }  \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{11} = \left( 
\begin{array}{rrrrr} 
 \frac{ 1 }{ 5 }  & 1 &  \frac{ 6 }{ 5 }  &  \frac{ 7 }{ 5 }  &  \frac{ 8 }{ 5 }  \\ 
1 & 0 &  - 4 &  - 8 &  - 12 \\ 
0 & 0 & 1 &  \frac{ 8 }{ 5 }  &  \frac{ 11 }{ 5 }  \\ 
0 & 0 & 0 & 1 &  \frac{ 17 }{ 11 }  \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; D_{11} = \left( 
\begin{array}{rrrrr} 
5 & 0 & 0 & 0 & 0 \\ 
0 &  -  \frac{ 1 }{ 5 }  & 0 & 0 & 0 \\ 
0 & 0 & 5 & 0 & 0 \\ 
0 & 0 & 0 &  \frac{ 11 }{ 5 }  & 0 \\ 
0 & 0 & 0 & 0 &  \frac{ 6 }{ 11 }  \\ 
\end{array}
\right) 
$$
==============================================
$$ P^T H P = D  $$
$$\left( 
\begin{array}{rrrrr} 
0 & 1 & 0 & 0 & 0 \\ 
1 &  -  \frac{ 1 }{ 5 }  & 0 & 0 & 0 \\ 
4 &  - 2 & 1 & 0 & 0 \\ 
 \frac{ 8 }{ 5 }  &  \frac{ 1 }{ 5 }  &  -  \frac{ 8 }{ 5 }  & 1 & 0 \\ 
 \frac{ 8 }{ 11 }  &  \frac{ 1 }{ 11 }  &  \frac{ 3 }{ 11 }  &  -  \frac{ 17 }{ 11 }  & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrrr} 
0 & 1 & 2 & 3 & 4 \\ 
1 & 5 & 6 & 7 & 8 \\ 
2 & 6 & 9 & 10 & 11 \\ 
3 & 7 & 10 & 12 & 13 \\ 
4 & 8 & 11 & 13 & 14 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrrr} 
0 & 1 & 4 &  \frac{ 8 }{ 5 }  &  \frac{ 8 }{ 11 }  \\ 
1 &  -  \frac{ 1 }{ 5 }  &  - 2 &  \frac{ 1 }{ 5 }  &  \frac{ 1 }{ 11 }  \\ 
0 & 0 & 1 &  -  \frac{ 8 }{ 5 }  &  \frac{ 3 }{ 11 }  \\ 
0 & 0 & 0 & 1 &  -  \frac{ 17 }{ 11 }  \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rrrrr} 
5 & 0 & 0 & 0 & 0 \\ 
0 &  -  \frac{ 1 }{ 5 }  & 0 & 0 & 0 \\ 
0 & 0 & 5 & 0 & 0 \\ 
0 & 0 & 0 &  \frac{ 11 }{ 5 }  & 0 \\ 
0 & 0 & 0 & 0 &  \frac{ 6 }{ 11 }  \\ 
\end{array}
\right) 
$$
$$ Q^T D Q = H  $$
$$\left( 
\begin{array}{rrrrr} 
 \frac{ 1 }{ 5 }  & 1 & 0 & 0 & 0 \\ 
1 & 0 & 0 & 0 & 0 \\ 
 \frac{ 6 }{ 5 }  &  - 4 & 1 & 0 & 0 \\ 
 \frac{ 7 }{ 5 }  &  - 8 &  \frac{ 8 }{ 5 }  & 1 & 0 \\ 
 \frac{ 8 }{ 5 }  &  - 12 &  \frac{ 11 }{ 5 }  &  \frac{ 17 }{ 11 }  & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrrr} 
5 & 0 & 0 & 0 & 0 \\ 
0 &  -  \frac{ 1 }{ 5 }  & 0 & 0 & 0 \\ 
0 & 0 & 5 & 0 & 0 \\ 
0 & 0 & 0 &  \frac{ 11 }{ 5 }  & 0 \\ 
0 & 0 & 0 & 0 &  \frac{ 6 }{ 11 }  \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrrr} 
 \frac{ 1 }{ 5 }  & 1 &  \frac{ 6 }{ 5 }  &  \frac{ 7 }{ 5 }  &  \frac{ 8 }{ 5 }  \\ 
1 & 0 &  - 4 &  - 8 &  - 12 \\ 
0 & 0 & 1 &  \frac{ 8 }{ 5 }  &  \frac{ 11 }{ 5 }  \\ 
0 & 0 & 0 & 1 &  \frac{ 17 }{ 11 }  \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rrrrr} 
0 & 1 & 2 & 3 & 4 \\ 
1 & 5 & 6 & 7 & 8 \\ 
2 & 6 & 9 & 10 & 11 \\ 
3 & 7 & 10 & 12 & 13 \\ 
4 & 8 & 11 & 13 & 14 \\ 
\end{array}
\right) 
$$