The chapter shows that the Kronecker product and the vec operator are intimately connected. It discusses the Moore㏄enrose (MP) inverse, which generalizes the concept of the inverse of a nonsingular matrix to singular square matrices and rectangular matrices. The inverse of a matrix is defined ...
Therefore, we mathematically propose a fast block-wise inverse Jacket transform of orders N = 2 k , 3 k , 5 k , and 6 k , where k is a positive integer. Based on the Kronecker product of the successive lower order Jacket matrices and the basis matrix, the fast algorithms for ...
where [Math Processing Error]δij is the Kronecker delta. Let us now denote [Math Processing Error]Ks=hKs(ti,tj), [Math Processing Error]Ks1=hKs1(ti,tj), [Math Processing Error]Ks2=hKs2(ti,tj), [Math Processing Error]Ks3=Ks3(ti). For a given vector v, the product Sv can then...
When tackling the abelian groups, it is useful to have a function field analogue of the Kronecker–Weber Theorem. To do this, we turn to Carlitz–Hayes theory [9]. For each polynomial M∈Fp[t] we define the Carlitz polynomial [M](x) with coefficients in Fp[t] recursively:[1](x)=x...
. . , m, where δi j is the Kronecker delta symbol. Let U := [u1, . . . , um] (26) be an n × m Krylov matrix with its j th column being the vector u j . Because u1, . . . , um are linearly independent vectors and m < n, the rank of U is rank(U) = m. ...
W can have different forms but the simplest one is based on the norm of the columns of the matrix G: W = Ω ⊗ I3, where ⊗ denotes the Kronecker product and Ω is a diagonal p× p matrix with Ω β β = ∑ α = 1 N g ( r α , r d i p β ) ⋅ g ( r α ,...
In this paper we analyze the decay pattern of the inverse of banded matrices of the form S=MIn+InM where M is tridiagonal, symmetric and positive definite, In is the identity matrix, and stands for the Kronecker product. It is well known that the inverses of banded matrices exhibit an ...
SUMMARYIn this paper, we investigate the normwise, mixed, and componentwise condition numbers and their upper bounds for the MoorePenrose inverse of the Kronecker product and more general matrix function compositions involving Kronecker products. We also present the condition numbers and their upper ...
The class of -matrices allows an approximate matrix arithmetic with almost linear complexity. In the present paper, we apply the -matrix technique combined with the Kronecker tensor-product approximation (cf. [2, 20]) to represent the inverse of a discrete elliptic operator in a hypercube (0,...
Kronecker productSeismic inversionQuasi-linear problemTensor decompositionInverse problems where relationships are linear arise in many fields of science and engineering and, consequently, algorithms for solving them are widespread. However, when the size of the problem increases, the computational challenge ...