Rotation properties of 2D isotropic dilation matrices

Abstract

In this paper, we consider the bivariate real isotropic dilation matrices that are similar (up to constant factors) to rotation matrices; and we show that, in this case, the two-scale relations can be considered also as relations between not only dilated but also rotated scaling functions. We present sufficient conditions on a matrix to be similar to a rotation matrix, where the change-of-basis matrix is symmetric positive definite. Also we present a simple test that the real matrix with integer entries performs rotation by an incommensurable to [Formula: see text] angle. We show that if a dilation matrix is similar to a rotation matrix, then an ellipse defined by the change-of-basis matrix is invariant (up to a uniform dilation) under transformation by the dilation matrix.