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A Sylvester equation has a unique solution for X exactly when there are no common eigenvalues of A and −B. More generally, the equation AX + XB = C has been considered as an equation of bounded operators on a (possibly infinite-dimensional) Banach space.
In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function f(A) of a matrix A as a polynomial in A, in terms of the eigenvalues and eigenvectors of A. [1] [2] It states that [3]
A generalization of Sylvester's construction proves that if and are Hadamard matrices of orders n and m respectively, then is a Hadamard matrix of order nm. This result is used to produce Hadamard matrices of higher order once those of smaller orders are known.
The rth emanant of a binary form in variables x i is a covariant given by the action of the rth power of the differential operator Σy i ∂/∂x i. This is essentially the same as polarization. (Elliott 1895, p.56) Sylvester (1853, Glossary p. 543–548) endoscopic See Sylvester (1853, Glossary p. 543–548). Archaic. equianharmonic contravariant
The clock matrix amounts to the exponential of position in a "clock" of hours, and the shift matrix is just the translation operator in that cyclic vector space, so the exponential of the momentum. They are (finite-dimensional) representations of the corresponding elements of the Weyl-Heisenberg group on a d {\displaystyle d} -dimensional ...
The United Nations Security Council is likely to meet on Monday, two diplomats said, over North Korea's test on Thursday of what Pyongyang said was an intercontinental ballistic missile. The ...
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For operators in a finite factor, one may define a positive real-valued determinant called the Fuglede−Kadison determinant using the canonical trace. In fact, corresponding to every tracial state on a von Neumann algebra there is a notion of Fuglede−Kadison determinant.