Similar matrices are two square matrices and that are related by

where is a square nonsingular matrix over the same field. A transformation of the form is called a similarity transformation, or conjugation by . Equivalently, is an invertible solution of the homogeneous Sylvester equation . For example,

and

are similar under conjugation by

Similar matrices represent the same linear transformation after a change of basis (for the domain and range simultaneously). Recall that a matrix corresponds to a linear transformation, and a linear transformation corresponds to a matrix after choosing a basis ,

Changing the basis changes the coefficients of the matrix,

If uses the standard basis vectors, then is the matrix using the basis vectors .

See also

Diagonal Matrix, Diagonalizable Matrix, Group, Jordan Canonical Form, Linear Transformation, Rational Canonical Form, Similarity Transformation, Square Matrix, Vector Basis, Vector Space

Portions of this entry contributed by Todd Rowland

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References

Golub, G. H. and Van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, p. 311, 1996.

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Similar Matrices

Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Similar Matrices." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SimilarMatrices.html

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