Case (1) corresponds to similarity transformations which generate a subgroup of similarities.

However, matrices that can be diagonalized with the same similarity transformation do form a commutative ring.

The geometric median is equivariant for Euclidean similarity transformations, including translation and rotation.

The similarity transformations form the subgroup where A is a scalar times an orthogonal matrix.

More simply, the similarity transformations can be seen to describe the growth and form of mollusc shells (Figure 3b).

An example is a shear, which changes two axes in different ways, or a similarity transformation, which preserves angles but not lengths.

While these similarity transformations capture some basic properties of plasmas, not all plasma phenomena scale in this way.

The proof of the main result of this section will rely on the similarity transformation as stated and proven next.

The matrix P is sometimes called a similarity transformation.

Likewise, we can make one ball into a larger or smaller ball by stretching, in other words, by applying similarity transformations.