In linear algebra, the Frobenius companion matrix of the monic polynomial p ( x ) = c 0 + c 1 x + ⋯ + c n − 1 x n − 1 + x n {\displaystyle p(x)=c_{0}+c_{1}x+\cdots...

&1&-c_{n-1}\end{bmatrix}}} This matrix is called the companion matrix of the polynomial p ( x ) {\displaystyle p(x)} . Companion matrix Krylov subspace PlanetMath:...

animal kept for companionship Companion parrot, a pet parrot that interacts with its owner Companion matrix, a matrix with a specific relation to its...

In linear algebra, a circulant matrix is a square matrix in which all rows are composed of the same elements and each row is rotated one element to the...

they are linearly independent. The matrix of the linear operator with respect to such a basis is the companion matrix of a monic polynomial; this polynomial...

to find all roots of a polynomial is to find the eigenvalues of the companion matrix of monic polynomial, which coincides with the roots of the polynomial...

algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It...

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems...

In mathematics, a matrix (pl.: matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows...

degree n {\displaystyle n} is the characteristic polynomial of some companion matrix of order n {\displaystyle n} .) Therefore, for matrices of order 5...

In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row:...

situation involves finding a minimal height companion matrix for the polynomial within a Bohemian matrix family. However, no efficient general-purpose...

Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily similar to an upper triangular matrix whose diagonal...

standard matrix form for K [ T ] / p ( T ) {\displaystyle K[T]/p(T)} , this yields various canonical forms: invariant factors + companion matrix yields...

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element...

{\displaystyle n\times n} companion matrix A = C ( G ) {\displaystyle A=C(G)} . r {\displaystyle r} steps are written as the matrix A r = C ( G ) r {\displaystyle...

field of rational functions, is generated by the formal variable X. Companion matrix for the multiplication map on a simple field extension (Roman 1995)...

matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries...

an − 1 xn − 1 + ⋯ + a0, then q(x) is the characteristic polynomial of the n×n companion matrix ( 0 0 ⋯ 0 − a 0 1 0 ⋯ 0 − a 1 0 1 ⋯ 0 − a 2 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ 1 − a n...

finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors. Given an n × n square matrix A of real or complex numbers...

matrices in F 2 {\displaystyle \mathbb {F} _{2}} (see GF(2)). Using the companion matrix of the characteristic polynomial of the LFSR and denoting the seed...

discrete wavelet transform with the discrete Fourier transform. Companion matrix DFT matrix Fast Fourier transform FFTPACK FFTW Generalizations of Pauli...

efficiently modulo n by computing Am mod n, where A is the corresponding k×k companion matrix. The above methods adapt easily to this application. This can be used...

Both bounds result from the Gershgorin circle theorem applied to the companion matrix of the polynomial and its transpose. They can also be proved by elementary...

square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the...

Every conjugate matrix T A T − 1 {\displaystyle TAT^{-1}} of A is as well a companion matrix of ƒ(X). Choosing T as diagonal matrix leaves the structure...

snapshot. Instead of computing the companion matrix S {\displaystyle S} , the SVD-based approach yields the matrix S ~ {\displaystyle {\tilde {S}}} that...

elementary application is obtaining the Frobenius normal form, i.e. the companion matrix of a polynomial. The commutative rings of split-complex numbers and...

also holds for matrices over the quaternions, a noncommutative ring. Companion matrix See Sect. 2 of Krivoruchenko (2016). An explicit expression for the...

Univariate Equation: Proxy Rootfinders, Chebyshev Interpolation, and the Companion Matrix". SIAM Review. 55 (2): 375–396. doi:10.1137/110838297. Scavo, T. R...