Direct and Inverse Transformations Between Phase Variable and Canonical Forms

1972 ◽  
Vol 94 (4) ◽  
pp. 315-318 ◽  
Author(s):  
R. R. Beck ◽  
G. M. Lance
Keyword(s):  
Canonical Form ◽  
General Treatment ◽  
Phase Variable ◽  
Canonical Forms ◽  
System Matrix ◽  
Jordan Canonical Form ◽  
Variable Form ◽  
Repeated Eigenvalues ◽  
Analytical Expressions ◽  
Real Complex

A complete, general treatment of the transformations, direct and inverse, between the phase variable form and the canonical or Jordan canonical form of the system matrix is presented. Analytical expressions are obtained for the matrices and all combinations of real, complex, distinct, and repeated eigenvalues are covered.

scholarly journals A Jordan canonical form for reachable linear systems

1989 ◽  
Vol 122-124 ◽  
pp. 489-524 ◽  
Author(s):  
D. Hinrichsen ◽  
D. Prätzel-Wolters
Keyword(s):  
Linear Systems ◽  
Canonical Form ◽  
Jordan Canonical Form

scholarly journals Spectra inhabiting the left half-plane that are universally realizable

2021 ◽  
Vol 10 (1) ◽  
pp. 180-192
Author(s):  
Ricardo L. Soto
Keyword(s):  
Canonical Form ◽  
Half Plane ◽  
New Left ◽  
Nonnegative Matrix ◽  
Complex Numbers ◽  
Jordan Canonical Form ◽  
Positivity Condition ◽  
Left Half ◽  
Left Half Plane ◽  
New Criteria

Abstract Let Λ = {λ1, λ2, . . ., λ n } be a list of complex numbers. Λ is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Minc ([21],1981) showed that if Λ is diagonalizably positively realizable, then Λ is universally realizable. The positivity condition is essential for the proof of Minc, and the question whether the result holds for nonnegative realizations has been open for almost forty years. Recently, two extensions of the Minc’s result have been proved in ([5], 2018) and ([12], 2020). In this work we characterize new left half-plane lists (λ1 > 0, Re λ i ≤ 0, i = 2, . . ., n) no positively realizable, which are universally realizable. We also show new criteria which allow to decide about the universal realizability of more general lists, extending in this way some previous results.

scholarly journals Reduced Canonical Forms of Stoppers

10.37236/1083 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Aaron N. Siegel
Keyword(s):  
Canonical Form ◽  
Canonical Forms ◽  
Alternate Proof

The reduced canonical form of a loopfree game $G$ is the simplest game infinitesimally close to $G$. Reduced canonical forms were introduced by Calistrate, and Grossman and Siegel provided an alternate proof of their existence. In this paper, we show that the Grossman–Siegel construction generalizes to find reduced canonical forms of certain loopy games.

Canonical forms for definable subsets of algebraically closed and real closed valued fields

1995 ◽  
Vol 60 (3) ◽  
pp. 843-860 ◽  
Author(s):  
Jan E. Holly
Keyword(s):  
Canonical Form ◽  
Simple Form ◽  
Finite Union ◽  
Canonical Forms ◽  
Boolean Combination ◽  
Valued Fields ◽  
The Real

AbstractWe present a canonical form for definable subsets of algebraically closed valued fields by means of decompositions into sets of a simple form, and do the same for definable subsets of real closed valued fields. Both cases involve discs, forming “Swiss cheeses” in the algebraically closed case, and cuts in the real closed case. As a step in the development, we give a proof for the fact that in “most” valued fields F, if f(x), g(x) ∈ F[x] and v is the valuation map, then the set {x: v(f(x)) ≤ v(g(x))} is a Boolean combination of discs; in fact, it is a finite union of Swiss cheeses. The development also depends on the introduction of “valued trees”, which we define formally.

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