# Any connections between a skew-symmetric matrix and a “positive reciprocal matrix”?

Consider a 7x7 matrix of the form

\begin{pmatrix} e&&B^{-1}\\ &\ddots\\ B&&e \end{pmatrix}

Where $e$ is a real number spanning the diagonal and $B$ is a 6x6 purely real matrix.

In a Skew-Symmetric Matrix (SSM), we would have

$e= 0$
$(B^{-1})_{ji}=-B_{ij}$ (by which I'm trying to say $A^T=-A$)

However, I'm working with a matrix in which

$e= 1$
$(B^{-1})_{ji}=1/B_{ij}$

To the best of my research, the only place I've encountered such a matrix is in Analytic Hierarchy Process (AHP) where it's known as a "Positive Reciprocal Matrix" (PRM). I've also come across something similar in statistics, where it's known as a "pairwise comparison matrix", though I've not come across a reciprocal version in that field.

Q1: Is there another field of study that uses a PRM?

Let's be specific:

To obtain my PRM, I put 7 data points (frequencies in units of Hz) as rows and the same 7 data points as columns and then took their ratios (hence the $e=1$ along the diagonal.)

\begin{pmatrix} 1.00& 1.33& 1.78& 1.19& 1.59& 1.06& 1.41 \\ 0.75& 1.00& 1.33& 0.89& 1.19& 0.79& 1.06 \\ 0.56& 0.75& 1.00& 0.67& 0.89& 0.59& 0.79 \\ 0.84& 1.12& 1.50& 1.00& 1.33& 0.89& 1.19 \\ 0.63& 0.84& 1.12& 0.75& 1.00& 0.67& 0.89 \\ 0.94& 1.26& 1.68 &1.12 &1.50 &1.00& 1.33 \\ 0.71& 0.94& 1.26& 0.84& 1.12& 0.75& 1.00 \\ \end{pmatrix}

Next, instead of taking their ratios, I took their difference (hence the $e=0$ along the diagonal) and generated a SSM from the same data
\begin{pmatrix} 0.000 &-123.890& -216.700& -78.580& -182.750& -27.720& -144.650 \\ 123.890 & 0.000 & -92.810 & 45.310 & -58.860 & 96.170& -20.760 \\ 216.700 & 92.810 & 0.000 & 138.120 & 33.950 &188.980& 72.050 \\ 78.580 &-45.310 &-138.120 & 0.000 &-104.170 & 50.860 & -66.070 \\ 182.750 & 58.860 & -33.950 &104.170 & 0.000 & 155.030 & 38.100 \\ 27.720 &-96.170 &-188.980 & -50.860& -155.030 & 0.000& -116.930 \\ 144.650 &20.760 & -72.050 & 66.070 & -38.100 & 116.930 & 0.000 \\ \end{pmatrix}

• Both the SSM and PRM have $determinant=0$.

• The SSM has $rank=2$ and $trace=0$ and the following eigenvalues

\begin{matrix} ( 0.000, 515.247i)\\ ( 0.000,-515.247i) \\ ( 0.000, 0.000i) \\ ( 0.000, 0.000i) \\ ( 0.000, 0.000i) \\ ( 0.000, 0.000i) \\ ( 0.000, 0.000i) \\ \end{matrix}

• The PRM has $rank=trace=7$ and the following eigenvalues
\begin{matrix} ( 6.994, 0.000i)\\ ( 0.006, 0.000i) \\ ( 0.002, 0.006i) \\ ( 0.002,-0.006i) \\ (-0.002, 0.000i) \\ (-0.001, 0.004i) \\ (-0.001,-0.004i) \\ \end{matrix}

• A Singular Value Decomposition, $USV^T$, on the SSM gave me the following S-matrix

\begin{pmatrix} 515.247 &0.000 &0.000 &0.000 &0.000 &0.000 &0.000\\ 0.000 &515.247 &0.000 &0.000 &0.000 &0.000 &0.000 \\ 0.000 &0.000 &0.000 &0.000 &0.000 &0.000 &0.000 \\ 0.000 &0.000 &0.000 &0.000 &0.000 &0.000 &0.000 \\ 0.000 &0.000 &0.000 &0.000 &0.000 &0.000 &0.000 \\ 0.000 &0.000 &0.000 &0.000 &0.000 &0.000 &0.000 \\ 0.000 &0.000 &0.000 &0.000 &0.000 &0.000 &0.000 \\ \end{pmatrix}

• A Singular Value Decomposition, $USV^T$, on the PRM gave me the following S-matrix

\begin{pmatrix} 7.535 &0.000 &0.000 &0.000 &0.000 &0.000 &0.000\\ 0.000 &0.009 &0.000 &0.000 &0.000 &0.000 &0.000\\ 0.000 &0.000 &0.007 &0.000 &0.000 &0.000 &0.000\\ 0.000 &0.000 &0.000 &0.006 &0.000 &0.000 &0.000\\ 0.000 &0.000 &0.000 &0.000 &0.005 &0.000 &0.000\\ 0.000 &0.000 &0.000 &0.000 &0.000 &0.003 &0.000\\ 0.000 &0.000 &0.000 &0.000 &0.000 &0.000 &0.002 \end{pmatrix}

• A LU decomposition on the SSM gave me the following U-matrix

\begin{pmatrix} 216.700 & 92.810 & 0.000 & 138.120 & 33.950 & 188.980 & 72.050 \\ 0.000& -123.890 &-216.700 &-78.580& -182.750 & -27.720& -144.650 \\ 0.000 &0.000 &0.000 &0.000 &0.000 &0.000 &0.000 \\ 0.000 &0.000 &0.000 &0.000 &0.000 &0.000 &0.000 \\ 0.000 &0.000 &0.000 &0.000 &0.000 &0.000 &0.000 \\ 0.000 &0.000 &0.000 &0.000 &0.000 &0.000 &0.000 \\ 0.000 &0.000 &0.000 &0.000 &0.000 &0.000 &0.000 \\ \end{pmatrix}

• A LU decomposition on the PRM gave the following U-matrix

\begin{pmatrix} 1.000 & 1.330 & 1.780 & 1.190 & 1.590 & 1.060 & 1.410 \\ 0.000 &0.010 &0.007 &0.001 &0.005 &0.004 &0.005 \\ 0.000 &0.000 &-0.007 &-0.003 &-0.004 &-0.006 &0.001 \\ 0.000 &0.000 &0.000 &-0.004 &-0.006 &0.000 &0.001 \\ 0.000 &0.000 &0.000 &0.000 -&0.008 &-0.005 &-0.002 \\ 0.000 &0.000 &0.000 &0.000 &0.000 &0.006 &0.001 \\ 0.000 &0.000 &0.000 &0.000 &0.000 &0.000 &0.006 \\ \end{pmatrix}

• A further property I uncovered is that for the PRM

$B^2=trace[B]B$

something that was discussed in this earlier question about a toy model for the PRM matrix. I see no easily discernible relationship for $B^2$ in the SSM case.

If there are other results that you need (like the permutation matrix of the LU or the other matrices of the $USV^T$), let me know so I can provide them.

I can already see some patterns in these results but I'm still unsure as to what they mean (I'm a physicist not a mathematician) which leads me to the following questions:

Q2: What is the best way to interpret all these results?

Q3: Do these results point to a relationship between the SSM and PRM that might help me understand my data better?

Q4: Are there any other properties to a PRM or SSM that would help me better understand the matrices, the data, or the structure I'm studying?

Finally, I can't help but notice a parallel between the basic forms of the SSM and PRM. In fact, if I interpret $e$ to be the identity, then it "seems" that the SSM is the "additive inverse" version (since the transpose elements are the negative of the original element) while the PRM is the "multiplicative inverse" version (since the transpose elements are the reciprocal of the original elements).

As well, we know that in the case of a SSM

$B^T = -B$

but even though the PRM transpose, given it's reciprocal nature, might be viewed as

$B^T = B^{-1}$

this cannot be the case for many reasons but foremost that it has $determinant =0$ and thus the matrix $B$ is non-invertible.

Since my investigation centers around additive groups, I can't shake the feeling that maybe ring theory is more suitable or that it can give me insight into what I'm studying.

Q5: Is there any validity to relating the PRM to a SSM?

If you are taking the logarithm of the elements, then you get a skew-symmetric matrix, since $$\ln 1 = 0$$ and $$\ln \left(1/a_{i,j}\right) = - \ln a_{i,j}.$$ By the way it is a known trick in some AHP-related method.