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Similar Matrices


Two square matrices A and B that are related by

 B=X^(-1)AX,
(1)

where X is a square nonsingular matrix are said to be similar. A transformation of the form X^(-1)AX is called a similarity transformation, or conjugation by X. For example,

 [0 1; 0 0]
(2)

and

 [0 0; 1 0]
(3)

are similar under conjugation by

 C=[0 1; 1 0].
(4)

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 b_i,

 T(sumlambda_ib_i)=suma_(ji)lambda_ib_j
(5)

Changing the basis changes the coefficients of the matrix,

 T(sumgamma_ie_i)=suma_(ji)^'gamma_ie_j
(6)

If T(v)=Av uses the standard basis vectors, then T is the matrix CAC^(-1) using the basis vectors b_i=Ce_i.


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.

Referenced on Wolfram|Alpha

Similar Matrices

Cite this as:

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

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