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It now follows from the properties of the Kronecker product that the equation AXB = C has a unique solution, if and only if A and B are invertible (Horn & Johnson 1991, Lemma 4.3.1). If X and C are row-ordered into the column vectors u and v, respectively, then (Jain 1989, 2.8 Block Matrices and Kronecker Products)
In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers. [1] The inequality is named after William Henry Young and should not be confused with Young's convolution inequality. Young's inequality for products can be used to prove Hölder's inequality.
A major unsolved problem in representation theory and combinatorics is to give a combinatorial description of the Kronecker coefficients. It has been open since 1938, when Murnaghan asked for such a combinatorial description. [2] A combinatorial description would also imply that the problem is # P-complete in light of the above result.
Kolmogorov's three-series theorem, combined with Kronecker's lemma, can be used to give a relatively easy proof of the Strong Law of Large Numbers. [ 1 ] Statement of the theorem
2 Tensor products of representations. ... 2.1 Kronecker coefficients. ... are called the cyclic exponents of with respect to the representation. [15 ...
where , is the inner product.Examples of inner products include the real and complex dot product; see the examples in inner product.Every inner product gives rise to a Euclidean norm, called the canonical or induced norm, where the norm of a vector is denoted and defined by ‖ ‖:= , , where , is always a non-negative real number (even if the inner product is complex-valued).
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 ...
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 graphs. Stochastic kronecker graph is a kronecker graph with each component of the matrix made by real numbers between 0 and 1. The stochastic version of kronecker graph eliminates the ...