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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]
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 ...
Sylvester’s rank inequality: if A is an m × n matrix and B is n × k, then [ii] + (). This is a special case of the next inequality. The inequality due to Frobenius : if AB , ABC and BC are defined, then [ iii ] rank ( A B ) + rank ( B C ) ≤ rank ( B ) + rank ( A B C ) . {\displaystyle \operatorname {rank} (AB ...
The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. The determinant of the Sylvester matrix of two polynomials is their resultant, which is zero when the two polynomials have a common root (in case of coefficients in a field) or a non-constant common divisor (in case of coefficients in an integral domain).
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.
PBX switchboard, 1975. A telephone switchboard is a device used to connect circuits of telephones to establish telephone calls between users or other switchboards. The switchboard is an essential component of a manual telephone exchange, and is operated by switchboard operators who use electrical cords or switches to establish the connections.
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