# Similar Matrices

## similar matrices

[¦sim·i·lər ′mā·tri‚sēz]
(mathematics)
Two square matrices A and B related by the transformation B = SAT, where S and T are nonsingular matrices and T is the inverse matrix of S.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## 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 λEAǀ and ǀλEBǀ and, consequently, the determinants ǀA and ǀBǀ and the eigenvalues of the similar matrices A and B coincide.

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(iii) Similar matrices: for commuting matrices [A.sub.1], ..., [A.sub.l] [member of] [GL.sub.n](k) and any S [member of] [GL.sub.n](k), we have D(S[A.sub.1][S.sup.-1], ..., S[A.sub.l][S.sup.-1]) = D([A.sub.1], ..., [A.sub.l]).
(ii) Similar matrices: ([A.sub.1], ..., [A.sub.l]) = (S[A.sub.1][S.sup.-1], ..., S[A.sub.l][S.sup.-1]) for commuting [A.sub.1], ..., [A.sub.l] [member of] [GL.sub.n](k) and any S [member of] [GL.sub.n](k).
These bubbles are the responsible of the tensile strength reductions for high temperatures and/or prolonged times described above, and they have been reported in the literature for similar matrices [22], Moreover, deformation lines are observed when curing temperature or time is increased, due to fracture in the material changes from "plastic" (Fig.
Now [D.sup.o] has the same eigenvalues as A(n) since they are similar matrices via S[D.sup.o] = A(n)S where S is upper triangular with entries
It is known that the matrix T(F* - KH*)[T.sup.-1] has the same eigenvalues [[lambda].sub.1], [[lambda].sub.2], and [[lambda].sub.3] as the one F* - KH* because they are similar matrices. One can obtain from (6) after simple transformations:
A new reference material (RM), RM 8504, has been prepared for use as a diluent oil with Aroclors in transformer oil Standard Reference Materials (SRMs) 3075 to 3080 and SRM 3090 when developing and validating methods for the determination of polychlorinated biphenyls (PCBs) as Aroclors in transformer oil or similar matrices. SRMs 3075-3080 and SRM 3090 consist of individual Aroclors in the same transformer oil that was used to prepare RM 8504.
RM 8504, Transformer Oil, is intended to be used as a diluent oil with transformer oil Standard Reference Materials (SRMs) 3075 to 3080 and SRM 3090 [1] when developing and validating methods for the determination of polychlorinated biphenyls (PCBs) as Aroclors (1) in transformer oil or similar matrices. This suite of Aroclor transformer oil SRMs consists of individual Aroclors in the same transformer oil that was used to prepare RM 8504 and is intended for use in the determination of PCBs in oil.
The QR algorithm with linear shift applied to the matrix A = [A.sub.0] defines a sequence of similar matrices according to the following rule:
Spatial tests such as Block Design do produce sex-specific results (Jensen 1980), but sex differences are not usually found with Raven's tests or similar matrices tests [see Table 4.33 in Kaufman and Kaufman (1983)].

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