Substitution Matrix

To a substitution $s$ with prototiles $T_1, \ldots T_m$ we assign the substitution matrix $M_s = (m_{ij})_{i,j = 1,\ldots ,m}$, where $m_{ij}$ is the number of copies of $T_i \in s(T_j)$. The substitution matrix carries a lot of information about the tilings arising from $s$. For simplicity, let’s assume the tilings are volume hierarchic. Then the eigenvector of $M_s$ which is largest in modulus is $q^d$; i.e., the inflation factor $q$ of $s$ to the dimension $d$. Note that this is true in any dimension $d$. If $M_s$ is primitive, then by the Perron-Frobenius theorem this largest eigenvalue is real, positive and unique. In this case, the corresponding (right) eigenvector contains the relative frequencies of the prototiles. Moreover, the left eigenvector (resp. the eigenvector of the transpose of $M_s$, if you don’t like left eigenvectors) contains the $d$-dimensional volumes of the prototiles. Further properties of the tilings can be derived from the algebraic properties of $q$, hence from the $d$-th root of the leading eigenvector of $M_s$ (for instance, see PV number).