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Let $\mu_{M,D}$ be the self-affine measure generated by an expanding integermatrix $M\in M_n(\mathbb{Z})$ and a finite digit set $D\subset\mathbb{Z}^n$. Itis well known that the two measures $\mu_{M,D}$ and $\mu_{\tilde{M},\tilde{D}}$have the same spectrality if $\tilde{M}=B^{-1}MB$ and $\tilde{D}=B^{-1}D$,where $B\in M_n(\mathbb{R})$ is a nonsingular matrix. This fact is usually usedto simplify the digit set $D$ or the expanding matrix $M$. However, it oftentransforms integer digit set $D$ or expanding matrix $M$ into real, whichbrings many difficulties to study the spectrality of$\mu_{\tilde{M},\tilde{D}}$. In this paper, we introduce a similaritytransformation of general linear group $GL_n(p)$ for some self-affine measures,and discuss their spectrality. This kind of similarity transformation can keepthe integer properties of $D$ and $M$ simultaneously, which leads to manyadvantages in discussing the spectrality of self-affine measures. As anapplication, we extend some well-known spectral self-affine measures to moregeneral forms.

The spectrality of self-affine measure under the similarity transformation of $GL_n(p)$