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.

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 It might be outdated or ideologically biased.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 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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References in periodicals archive

(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).

A Topology on Milnor's Group of a Topological Field and Continuous Joint Determinants

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.

Mechanical and interfacial properties of phenolic composites reinforced with treated cellulose fibers

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

A test matrix for an inverse eigenvalue problem

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:

Robust fault diagnosis in linear systems

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.

Preparation of reference material 8504, transformer 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:

Improved initialization of the accelerated and robust QR-like polynomial root-finding

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)].

Critique of Vreugdenhil et al.'s study linking PCBs to the play behaviors of Dutch girls and boys. (Correspondence)

With the CD-MAP, similar matrices are deployed to analyse the needs of internal customers and stakeholders using the same techniques as for external customers.

The consumer association: one reason why traditional approaches to business planning often fail to deliver is that companies aren't nearly as customer-focused as they'd like to think. Mike Walker ventures beyond the balanced scorecard with customer-driven planning and activity-based costing. (Finance Customer-Driven Planning)