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Similar Matrices
(redirected from Similar matrix)

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similar matrices [¦sim·i·lər ′mā·tri‚sēz]
(mathematics)
Two square matricesAandBrelated by the transformationB=SAT, whereSandTare nonsingular matrices andTis the inverse matrix ofS.
Similar Matrices 

Two square matrices A and B of order n are said to be similar if there exists a nonsingular, or invertible, matrix P of order n such that B= P-1AP. Similar matrices are obtained when the matrix of a linear transformation is given in different coordinate systems. The role of the matrix P in this case is played by the matrix of the transformation of coordinates. For a given matrix A it is often important to select a second matrix B that is similar to A and has as simple a form as possible—for example, the Jordan matrix. Similar matrices are of identical rank. The characteristic polynomials ǀ λE – Aǀ and ǀλE – Bǀ and, consequently, the determinants ǀA and ǀBǀ and the eigenvalues of the similar matrices A and B coincide.
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According to this report, resonant frequency or ultrasonic velocity measurements can be used to predict the effect of small changes in graphite form on the tensile strength, offset yield strength and other properties of ductile iron castings of known similar matrix structure containing up to 70% non-nodular graphite.
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