Kronecker


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Kro·nec·ker

(krō'nek-ĕr),
Karl H., Swiss physiologist, 1839-1914. See: Kronecker stain.
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NOMENCLATURE k scan time index j system model index [[bar.x].sub.k] system state vector [y.sub.k] measurement vector [A.sub.j] state transition matrix [B.sub.j] input matrix f(*) measurement model [R.sup.j] process noise covariance matrix [M.sup.j] measurement noise covariance matrix N covariance matrix of [n.sub.k] S set of all models [v.sup.j.sub.k] process noise vector [[omega].sup.j.sub.k] measurement noise vector [[delta].sub.k(k-i)] Kronecker delta function [[??].sup.j.sub.k] system model at kth step [[pi].sub.ij] transitions probability from ith model to jth model [phi] initial state distribution of Markov chain [[mu].sub.k] adversary steering command vector [[mu].sup.j.sub.k] mode probability Digital Object Identifier10.4316/AECE.2017.03005
The Kronecker endomorphism I [member of] [GAMMA](End([pi])) = End([GAMMA]([pi])) acts locally as:
If the D-dimensional sample signal X = [[x.sup.T.sub.1], [x.sup.T.sub.2], ..., [x.sup.T.sub.D]] and independent measurements result Y = [[y.sup.T.sub.1], [y.sup.T.sub.2], ..., [y.sup.T.sub.D]] are unknown, the Kronecker product measurement matrix [5] can be expressed as [bar.[PHI]] = [[PHI].sub.1] [cross product] [[PHI].sub.2] [cross product] ...
Huang, "Novel synchronization analysis for complex networks with hybrid coupling by handling multitude Kronecker product terms," Neurocomputing, vol.
The reader is referred to [23] for other properties of the Kronecker product not mentioned here.
The robust stabilization in the paper [9] (which published the same results as can be found here) was based on the similar idea as in the contribution [5], but it used the alternative methods, namely the combination of the Kronecker summation method [10] and the algebraic approach to control design under the ring of proper and stable rational functions ([R.sub.PS]) [11], [12], [13], [14], [15].
where [G.sub.d] is the (co)variance matrix of random direct additive genetic effects; [G.sub.m] is the (co) variance matrix of random maternal additive genetic effects; Q is the (co)variance matrix of random permanent maternal environment effects; R is the (co)variance matrix of random residual effects, A is the numerator relationship matrix, [I.sub.v] is the identity matrix with order v, where v is the number of mothers, [I.sub.n] is the identity matrix with order n, where n is the number of observations, and [R] is the Kronecker product operator.
[f.sub.m]([x.sub.n]) = [[delta].sub.m,n] (the Kronecker symbol) m, n [member of] N.
G is considered as the genetic covariance matrix of the random regression coefficients assumed to be the same for all cows; A is considered as the additive genetic relationship matrix among all animals; is the Kronecker product operator; P is the permanent environment covariance matrix of the random regression coefficients assumed to be the same for all cows; residual covariance matrix was equal to (Eq.) in which I is an identity matrix, and the residual variance was assumed to be constant throughout the lactation.
In case if the users equipped with the same number of receive antennas, then the combining strategy is simply the Kronecker product between [T.sup.(i).sub.u] and [T.sup.(j).sub.sk].
In the twelfth chapter on Graph Generation by Kronecker Multiplication, the authors have provided a brief description to offset the unequal distribution of adjacency matrix followed by a summary in the next chapter.