Example. Exact covariance on the alternating four-cycle [fdip-0037]
AGENTDRAFTED
Take the four-cycle on \(V=(0,1,2,3)\), with forward
edges and weights \((1,t,1,t)\), \(t>0\). Use counting pairing and
\(Q=L+m^2I\), \(m>0\). Put \(a=m^2\) and \(s=1+t+a\).
The following functions on vertices form an orthonormal eigenbasis;
the middle column lists the unnormalized rows \(2u_i^T\):
\[\begin {array}{c|c|c}
i&2u_i^T&\lambda _i\\\hline
0&(1,1,1,1)&0\\
1&(1,-1,-1,1)&2\\
2&(1,1,-1,-1)&2t\\
3&(1,-1,1,-1)&2+2t
\end {array}\]
Set
\[\begin {aligned}
q_0&=a,\\
q_1&=a+2,\\
q_2&=a+2t,\\
q_3&=a+2+2t,\\
\Delta &=a(a+2)(a+2t)(a+2+2t).
\end {aligned}\]
All 4 \(q_i\) are positive. With \(P_{u_i}=u_i u_i^T\), normalization and
inverse covariance give
\[\begin {aligned}
\det Q&=\Delta ,& Z_Q&=4\pi ^2/\sqrt \Delta ,\\
G&=\sum _{i=0}^3q_i^{-1}P_{u_i}.
\end {aligned}\]
With \(O\) the array whose columns are \(u_i\), this is
\(G=O\operatorname {diag}(q_i^{-1})O^T\).
Write \(d_i=q_i^{-1}\). Multiplying the signs of the relevant
coordinates in each projector gives
\[\begin {aligned}
4G_{00}&=d_0+d_1+d_2+d_3,\\
4G_{01}&=d_0-d_1+d_2-d_3,\\
4G_{02}&=d_0-d_1-d_2+d_3,\\
4G_{03}&=d_0+d_1-d_2-d_3.
\end {aligned}\]
Thus the complete covariance, with \((A,B,C,D)=(G_{00},G_{01},G_{02},G_{03})\), is
\[G=\begin {pmatrix}
A&B&C&D\\B&A&D&C\\C&D&A&B\\D&C&B&A
\end {pmatrix}.\]
In particular the variance and adjacent two-point expectation are the
first two formulas, with their indicated signs.
For the opposite entry, the common denominator \(4\Delta \) has
numerator
\[\begin {aligned}
N&=(a+2)(a+2t)(a+2+2t)\\
&\quad -a(a+2t)(a+2+2t)\\
&\quad -a(a+2)(a+2+2t)\\
&\quad +a(a+2)(a+2t)\\
&=2[(a+2t)(a+2+2t)-a(a+2)]\\
&=8t(a+1+t).
\end {aligned}\]
Consequently \(G_{02}=2ts/\Delta \). At \((t,m)=(1,1)\),
\(q=(1,3,3,5)\), \(\Delta =45\), \(Z_Q=4\pi ^2/(3\sqrt 5)\), and
\[\begin {aligned}
&(G_{00},G_{01},G_{02},G_{03})\\
&\qquad =(7/15,1/5,2/15,1/5).
\end {aligned}\]
At \((t,m)=(2,1)\), \(q=(1,3,5,7)\), \(\Delta =105\), and
\[(G_{00},G_{01},G_{02},G_{03})
=(44,19,16,26)/105.\]
These numbers are obtained by direct substitution in the 4 signed
sums above. They give, in particular, the opposite covariance used in
block elimination and the two distinct adjacent covariances
used in relabeling.
More generally, weights \((b,c,b,c)\) with \(b,c>0\) keep the same
4 eigenvectors and give eigenvalues \(0,2b,2c,2b+2c\), by the same
coordinate products. This is a real spectral calculation. The
arithmetic columns are \(2u_i\), whose Gram matrix is \(4I\);
they are not an orthonormal basis over an arbitrary coefficient ring.
The eigenvectors and signed sums are computed here from the weighted
cycle.
[boyd2004convex, Section A.5.2, pp. 646-647] gives general
spectral decomposition.
[friedli2017statistical, Proposition B.55, p. 505] gives the
Gaussian density. Covariance equals inverse precision by Theorem [fdip-000B].