Limits for the characteristic roots of a matrix (I)

1968 ◽  
pp. 21-28
Author(s):  
Geng Ji (Keng Chi) [G. Gun]
Keyword(s):  
Characteristic Roots

Further Results on GGD Designs

1982 ◽  
Vol 31 (1-2) ◽  
pp. 53-62
Author(s):  
Sanjeev C. Panandikar
Keyword(s):  
New Methods ◽  
Characteristic Roots ◽  
Group Divisible ◽  
The Matrix

In this paper we discuss some of the properties of the matrix ( NN') of the Generalised Group Divisible (GGD) design with λ ij ; i, j=l,2. The properties are the characteristic roots, the Hasse­Minkowski invariant and the nonexistence theorems. Some new methods for constructing GGD designs are also given.

A simulation study of nonideal mixing effect on the dynamic response of an exothermic CSTR with Cholette’s model

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Chane-Yuan Yang ◽  
Yu-Shu Chien ◽  
Jun-Hong Chou
Keyword(s):  
Transfer Function ◽  
Dynamic Response ◽  
The Other ◽  
System Stability ◽  
Mixing Effect ◽  
Characteristic Roots ◽  
Mixing Parameter ◽  
Perfect Mixing ◽  
And Control ◽  
Inverse Response

Abstract The study of nonideal mixing effect on the dynamic behaviors of CSTRs has very rarely been published in the literature. In this work, Cholette’s model is employed to explore the nonideal mixing effect on the dynamic response of a nonisothermal CSTR. The analysis shows that the mixing parameter n (the fraction of the feed entering the zone of perfect mixing) and m (the fraction of the total volume of the reactor), indeed affect the characteristic roots of transfer function of a real CSTR, which determine the system stability. On the other hand, the inverse response and overshoot response are also affected by the nonideal mixing in a nonisothemal CSTR. These results are of much help for the design and control of a real CSTR.

Analytic functions of finite dimensional linear transformations

1959 ◽  
Vol 55 (1) ◽  
pp. 51-61 ◽  
Author(s):  
S. N. Afriat
Keyword(s):  
Power Series ◽  
Integral Formula ◽  
Characteristic Root ◽  
Interpolation Formula ◽  
General Function ◽  
Linear Transformations ◽  
Multiple Characteristic ◽  
Characteristic Roots ◽  
Image Position ◽  
Necessary And Sufficient

Since the first introduction of the concept of a matrix, questions about functions of matrices have had the attention of many writers, starting with Cayley(i) in 1858, and Laguerre(2) in 1867. In 1883, Sylvester(3) defined a general function φ(a) of a matrix a with simple characteristic roots, by use of Lagrange's interpolation formula, and Buchheim (4), in 1886, extended his definition to the case of multiple characteristic roots. Then Weyr(5) showed in 1887 that, for a matrix a with characteristic roots lying inside the circle of convergence of a power series φ(ζ), the power series φ(a) is convergent; and in 1900 Poincaré (6) obtained the formulaefor the sum, where C is a circle lying in and concentric with the circle of convergence, and containing all the characteristic roots in its ulterior, such a formula having effectively been suggested by Frobenius(7) in 1896 for defining a general function of a matrix. Phillips (8), in 1919, discovered the analogue, for power series in matrices, of Taylor's theorem. In 1926 Hensel(9) completed the result of Weyr by showing that a necessary and sufficient condition for the convergence of φ(a) is the convergence of the derived series φ(r)(α) (0 ≼ r < mα; α) at each characteristic root α of a, of order r at most the multiplicity mα of α. In 1928 Giorgi(10) gave a definition, depending on the classical canonical decomposition of a matrix, which is equivalent to the contour integral formula, and Fantappie (11) developed the theory of this formula, and obtained the expressionfor the characteristic projectors.

Discriminance of characteristic roots area for interval systems

1999 ◽  
Vol 32 (2) ◽  
pp. 3213-3218
Author(s):  
Yoshifumi Okuyama ◽  
Fumiaki Takemori ◽  
Hong Chen
Keyword(s):  
Characteristic Roots ◽  
Interval Systems

scholarly journals The Expected Characteristic and Permanental Polynomials of the Random Gram Matrix

10.37236/3709 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Jacob G. Martin ◽  
E. Rodney Canfield
Keyword(s):  
Random Matrix ◽  
Generating Functions ◽  
Numerical Algorithms ◽  
Singular Values ◽  
Gram Matrix ◽  
Efficient Computation ◽  
Underlying Distribution ◽  
Characteristic Roots ◽  
Speed Up ◽  
Theoretical Results

A $t \times n$ random matrix $A$ can be formed by sampling $n$ independent random column vectors, each containing $t$ components. The random Gram matrix of size $n$, $G_{n}=A^{T}A$, contains the dot products between all pairs of column vectors in the randomly generated matrix $A$, and has characteristic roots coinciding with the singular values of $A$. Furthermore, the sequences $\det{(G_{i})}$ and $\text{perm}(G_{i})$ (for $i = 0, 1, \dots, n$) are factors that comprise the expected coefficients of the characteristic and permanental polynomials of $G_{n}$. We prove theorems that relate the generating functions and recursions for the traces of matrix powers, expected characteristic coefficients, expected determinants $E(\det{(G_{n})})$, and expected permanents $E(\text{perm}(G_{n}))$ in terms of each other. Using the derived recursions, we exhibit the efficient computation of the expected determinant and expected permanent of a random Gram matrix $G_{n}$, formed according to any underlying distribution. These theoretical results may be used both to speed up numerical algorithms and to investigate the numerical properties of the expected characteristic and permanental coefficients of any matrix comprised of independently sampled columns.

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