The term "similarity transformation" is used either to refer to a geometric similarity, or to a transformation of a square matrix into a similar matrix.

A geometric similarity preserves angles and multiplies all distances by the same positive factor. It transforms objects in space to similar objects. Geometric similarities and the concept of self-similarity are important foundations of fractals and iterated function systems.

For square matrices, a similarity transformation has the form

where is an invertible matrix of the same size and over the same field as . The matrices and are called similar matrices (Golub and Van Loan 1996, p. 311). This algebraic transformation need not be a geometric similarity.

The determinant of the similarity transformation of a matrix is equal to the determinant of the original matrix

The determinant after subtracting a multiple of the identity matrix is given by

If is an antisymmetric matrix () and is an orthogonal matrix (), then the matrix for the similarity transformation

is itself antisymmetric, i.e., , where is the transpose. This follows using index notation for matrix multiplication, which gives

Here, the first equality follows from the definition of matrix multiplication, and the second uses the properties of the antisymmetric matrix and the orthogonal matrix . The third equality rearranges the factors using the commutative property of multiplication of scalars. The final equality follows again from the definition of matrix multiplication.

The similarity transformation of a subgroup of a group by any fixed group element in gives the conjugate subgroup (Arfken 1985, p. 242).

See also

Affine Transformation, Conformal Mapping, Determinant, Dilation, Iterated Function System, Normal Subgroup, Similar Matrices, Similarity

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 3, 1991.Golub, G. H. and Van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, p. 311, 1996.Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 83-103, 1991.

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Similarity Transformation

Cite this as:

Weisstein, Eric W. "Similarity Transformation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SimilarityTransformation.html

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