Similarity (geometry)

Informally, two objects are similar if they are similar in every aspect except possibly size or orientation. For example, a globe and the surface of the earth are, in theory, similar.

More formally, we say two objects are congruent if they are the same up to translation, rotation and reflection (rigid motions). We say two objects are similar if they are congruent up to a dilation.

All circles are similar. Two triangles are similar if they have the same angles (AAA similarity), and since the sum of the angles of any triangle is 180 degrees, this means that two triangles are similar if they have two equal angles (AA similarity). Equivalently, two triangles are similar if their corresponding sides are in equal ratios. Two polygons are similar if their corresponding angles are equal and corresponding sides are in a fixed ratio. Note that for polygons with 4 or more sides, both of these conditions are necessary. For instance, all rectangles have the same angles, but not all rectangles are similar.

Linear algebra

In linear algebra, two square matrices are similar if there exists an unitary matrix such that .

If has distinct eigenvalues, then it has a basis of eigenvectors and will be similar to a diagonal matrix.

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