similar matrix


A square matrixMathworldPlanetmath A is similarMathworldPlanetmathPlanetmathPlanetmath (or conjugatePlanetmathPlanetmath) to a square matrix B if there exists a nonsingular square matrix S such that

A=S-1BS. (1)

Note that, given S as above, we can define R=S-1 and have A=RBR-1. Thus, whether the inversePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath comes first or last does not matter.

TransformationsMathworldPlanetmath of the form S-1BS (or SBS-1) are called similarity transformations.


Similarity is useful for turning recalcitrant matrices into pliant ones. The canonical example is that a diagonalizable matrixMathworldPlanetmath A is similar to the diagonal matrixMathworldPlanetmath of its eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath Λ, with the matrix of its eigenvectorsMathworldPlanetmathPlanetmathPlanetmath acting as the similarity transformation. That is,

A =TΛT-1 (2)
=[v1v2vn][λ100λ2λn][v1v2vn]-1. (3)

This follows directly from the equation defining eigenvalues and eigenvectors,

AT=TΛ. (4)

If A is symmetricMathworldPlanetmathPlanetmathPlanetmathPlanetmath ( for example, then through this transformation, we have turned A into the productPlanetmathPlanetmath of two orthogonal matricesMathworldPlanetmath and a diagonal matrix. This can be very useful. As an application, see the solution for the normalizing constant of a multidimensional Gaussian integral.

Properties of similar matrices

  1. 1.

    Similarity is reflexiveMathworldPlanetmathPlanetmathPlanetmath ( All square matrices A are similar to themselves via the similarity transformation A=I-1AI, where I is the identity matrixMathworldPlanetmath with the same dimensionsPlanetmathPlanetmath as A.

  2. 2.

    Similarity is symmetric ( If A is similar to B, then B is similar to A, as we can define a matrix R=S-1 and have

    B=R-1AR (5)
  3. 3.

    Similarity is transitiveMathworldPlanetmathPlanetmathPlanetmath ( If A is similar to B, which is similar to C, we have

    A=S-1BS=S-1(R-1CR)S=(S-1R-1)C(RS)=(RS)-1C(RS). (6)
  4. 4.

    Because of 1, 2 and 3, similarity defines an equivalence relationMathworldPlanetmath () on square matrices, partitioning ( the space of such matrices into a disjoint set of equivalence classesMathworldPlanetmath.

  5. 5.

    If A is similar to B, then their determinantsMathworldPlanetmath are equal; i.e. (, detA=detB. This is easily verified:

    detA=det(S-1BS)=det(S-1)detBdetS=(detS)-1detBdetS=detB. (7)

    In fact, similar matrices have the same characteristic polynomialMathworldPlanetmathPlanetmath, which implies this result directly, the determinant being the constant term of the characteristic polynomial (up to sign).

  6. 6.

    Similar matrices represent the same linear transformation after a change of basis.

  7. 7.

    It can be shown that a matrix A and its transposeMathworldPlanetmath AT are always similar.

Title similar matrix
Canonical name SimilarMatrix
Date of creation 2013-03-22 12:24:37
Last modified on 2013-03-22 12:24:37
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 19
Author Wkbj79 (1863)
Entry type Definition
Classification msc 15A03
Synonym similarity transformation
Synonym similar matrices
Synonym conjugate matrices
Related topic Eigenvalue
Related topic Eigenvector
Related topic EigenvalueProblem
Defines similar
Defines conjugate