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In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix.It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis.
and the corresponding operation of symmetric functions is the usual product. Also note that the Littlewood–Richardson coefficients are the analogue of the Kronecker coefficients for representations of GL n, i.e. if we write W λ for the irreducible representation corresponding to λ (where λ has at most n parts), one gets that
Kronecker graphs are a construction for generating graphs for modeling systems. The method constructs a sequence of graphs from a small base graph by iterating the Kronecker product. [1] A variety of generalizations of Kronecker graphs exist. [2] The Graph500 benchmark for supercomputers is based on the use of a stochastic version of Kronecker ...
The Golden–Thompson inequality can be viewed as a generalization of a stronger statement for real numbers. If a and b are two real numbers, then the exponential of a+b is the product of the exponential of a with the exponential of b:
Matrix exponential; Matrix mortality problem; Matrix multiplication; Frobenius inner product; Matrix multiplication algorithm; Matrix polynomial; Matrix ring; Matrix semialgebra; Matrix semiring; Matrix sign function; Minimal polynomial (linear algebra) Minimum degree algorithm; Minor (linear algebra) Moore–Penrose inverse; Mutual coherence ...
The Kronecker delta has the so-called sifting property that for : = =. and if the integers are viewed as a measure space, endowed with the counting measure, then this property coincides with the defining property of the Dirac delta function () = (), and in fact Dirac's delta was named after the Kronecker delta because of this analogous property ...
Kronecker substitution is a technique named after Leopold Kronecker for determining the coefficients of an unknown polynomial by evaluating it at a single value. If p(x) is a polynomial with integer coefficients, and x is chosen to be both a power of two and larger in magnitude than any of the coefficients of p, then the coefficients of each term of can be read directly out of the binary ...
By taking Kronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large j , can be calculated using this spin operator and ladder operators .