Solutions of the Matrix Equation A
 *
=XA
=AX

“…Theorem 1 Let A ∈ R n×n , b ∈ R n×1 and c ∈ R 1×n be some matrices, the characteristic polynomial of matrix A be α(s) defined in (1) and the transfer function of (A, b, c) be defined in (5). Then…”

Section: The Main Theoremmentioning

confidence: 99%

An identity concerning controllability observability and coprimeness of linear systems and its applications

Zhou

Duan

Song

2007

J. Control Theory Appl.

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It is shown in this paper that any state space realization (A, b, c) of a given transfer function T (s) = β(s) α(s) with α(s) monic and dim(A) = deg(α(s)), satisfies the identity β(A) = Qc(A, b)SαQo(A, c) where Qc(A, b) andQo (A, c) are the controllability matrix and observability matrix of the matrix triple (A, b, c), respectively, and Sα is a nonsingular symmetric matrix. Such an identity gives a deep relationship between the state space description and the transfer function description of single-input single-output (SISO) linear systems. As a direct conclusion, we arrive at the well-known result that a realization of any transfer function is minimal if and only if the numerator and the denominator of the transfer function is coprime. Such a result is also extended to the SISO descriptor linear system case. As an applications, a complete solution to the commuting matrix equation AX = XA is proposed and the minimal realization of multi-input multi-output (MIMO) linear system is considered.

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“…EXAMPLE 1: Matrix satisfying (1) but not (2). Let the base field be GF (7) with the identity automorphism. (Example taken from reference [9].)…”

Section: The Structure Of Normal Matricesmentioning

confidence: 99%

“…wherep(x) is the polynomial which transforms A intoA*. Evaluatingeq (5)(6)(7)(8) atx=A the following is obtained: q(p(A)) = pr(O)q(A) . (5)(6)(7)(8)(9) Define the matrix B as follows:…”

Section: Qedmentioning

confidence: 99%

Normal matrices with entries from an arbitrary field of characteristic not-equal 2

Pearl¹,

Penn²

1972

J. RES. NATL. BUR. STAN. SECT. B. MATH. SCI.

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Le t F be a n a rbitrary fi e ld of c ha racte ri sti c 7" 2 a nd le t co nj ugati on in F be de fin e d b y a n a rbitra ry involutory a uto morphi s m (poss ibl y th e ide ntit y ma ppin g). A matrix with e ntri es fro m F is norm al if it co mmut es with its co nju gate tra ns pose. Seve ra l co nditi o n whi c h a re equi va le nt to norm alit y wh e n F is th e co mpl ex fi e ld are pro pe rl y nes te d whe n F is a n a "b itrary fi e ld. Of th ese con diti ons, no rm a lit y is th e wea kes t a nd unit a ry di ago nali zabi lit y is th e s tronges t. Wh e n th e unde rl yin g fi e ld is c losed th e unit ari ly dia go na lizabl e matri ces a re th ose whi c h possess a s pectra l re prese nt a ti o n with He rmiti a n ide mpote nt s. These matri ces may be ch arac teri ze d in ter ms of th eir EP pro pe rti es. A m atri x is ind ecomposa bl e if it has a s in gle e le mc nt a ry di viso r of the form (X-S)i. For i > 1, normal indecomposable matrices do not exist whe n F is the c ompl ex fi eld. H o we ve r th ere exi st fi elds for wh ic h normal indecomposa bl e ma tri ces of all finit e orde rs ex is t. A matri x is a no rm al indeco mposa ble matri x if a nd onl y if it ca n be ex pressed as r(B) wh e re r(x) is a polyno mi a l s uc h th at r'(O) 7" 0 a nd B is a ma trix ha vin g th e sin gle e le me nt a ry divi~o r Xi and sati sfyin g th e e qu ati o n B*= sB [o r so m e sca la r s. Wh e n th e in vo lut ory a utomorphi s m o[ F is th e id e ntit y m a p pin g in deco mposable nor ma l matri ces of e ve n o rder ex ist if a nd o nl y if th e vec to r s pace F" is hype rb oli c , and in thi s case th e ma tri ces a re sy mm e tri c. Moreov e r , ind eco mposabl e no rm al matri ces of odd o rd e r ex is t if an d "nl y if F " is th e o rth ogo na l SUm o[ a hype rb oli c s pace a nd a o ne dim e ns iona l s pace. a nd in thi s case both sy mm e tric an d no nsymme tri c ind eco mposabl e norm al ma t ri ces exis t.

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Solutions of the Matrix Equation A ^* =XA =AX

Cited by 5 publications

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Factorizations of EP operators

Factorizations of EP operators

An identity concerning controllability observability and coprimeness of linear systems and its applications

Normal matrices with entries from an arbitrary field of characteristic not-equal 2

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Solutions of the Matrix Equation A * =XA =AX

Cited by 5 publications

References 0 publications

Factorizations of EP operators

Factorizations of EP operators

An identity concerning controllability observability and coprimeness of linear systems and its applications

Normal matrices with entries from an arbitrary field of characteristic not-equal 2

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Product

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Solutions of the Matrix Equation A ^* =XA =AX