I have a matrix that is “kind of symmetric.” Specifically, it is an $n \times n$ real matrix such that the entries $a_{ij}=1/a_{ji}$ whenever $j \ne i$. I want to investigate the properties of this matrix. As a starting place, does this kind of matrix (or perhaps the matrix where $a_{ii}=1$ as well) have a special name and has it been studied?
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12$\begingroup$ I would say it is more “kind of skew-symmetric” than “kind of symmetric.” In fact, I might call this matrix “multiplicatively skew-symmetric” if another name doesn’t already exist… EDIT: indeed, Google confirms that “multiplicatively skew-symmetric” has been used for this property. $\endgroup$– Sam HopkinsCommented Dec 27, 2023 at 3:54
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2$\begingroup$ Skew symmetric matrices are interesting largely because they are the tangent space to the identity in the orthogonal group. I'd hesitate to call these matrices skew symmetric in any sense unless there was some kind of similar interesting interpretation to this symmetry. $\endgroup$– Ryan BudneyCommented Dec 27, 2023 at 4:59
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2 Answers
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The name of an $n\times n$ matrix with positive real elements satisfying $a_{ij}=1/a_{ji}$ for all $i,j\in\{1,2,\ldots n\}$ is reciprocal matrix.
A consistent reciprocal matrix has elements of the form $a_{ij}=w_i/w_j$, so that $a_{ij}a_{jk}=a_{ik}$. Its largest eigenvalue is equal to $n$ (here is a proof)
Consistent reciprocal matrices have applications in decision theory.
answered Dec 27, 2023 at 10:28
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Some call them currency exchange matrices. From Boyd & Vandenberghe's Introduction to Applied Linear Algebra:
6.7 Currency exchange matrix. We consider a set of $n$ currencies, labeled $1,\dots,n$. (These might correspond to USD, RMB, EUR, and so on.) At a particular time the exchange or conversion rates among the $n$ currencies are given by an $n \times n$ (exchange rate) matrix $R$, where $R_{ij}$ is the amount of currency $i$ that you can buy for one unit of currency $j$. (All entries of $R$ are positive.) The exchange rates include commission charges, so we have $R_{ji} R_{ij} < 1$ for all $i \neq j$. You can assume that $R_{ii} = 1$. Suppose $y = Rx$, where $x$ is a vector (with nonnegative entries) that represents the amounts of the currencies that we hold. What is $y_i$? Your answer should be in English.
Note that you are considering currency exchange matrices where $R_{ij} R_{ji} = 1$, which one could call perfect currency exchange matrices. I would have written $R_{ij} R_{ji} \leq 1$ instead, in order to include the (reciprocal) perfect currency exchange matrices, too.