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A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring M n (R). Matrix polynomials are often demonstrated in undergraduate linear algebra classes due to their relevance in showcasing properties of linear transformations represented as matrices, most notably the Cayley–Hamilton ...
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]
Download as PDF; Printable version; ... a Sylvester equation is a matrix equation of the form: [1] ... for any polynomial . In particular, let be the ...
A polynomial matrix over a field with determinant equal to a non-zero element of that field is called unimodular, and has an inverse that is also a polynomial matrix. Note that the only scalar unimodular polynomials are polynomials of degree 0 – nonzero constants, because an inverse of an arbitrary polynomial of higher degree is a rational function.
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.
Therefore the polynomial equation p A (λ) = 0 has at most n different solutions, that is, eigenvalues of the matrix. [42] They may be complex even if the entries of A are real. According to the Cayley–Hamilton theorem, p A (A) = 0, that is, the result of substituting the matrix itself into its characteristic polynomial yields the zero matrix.
An m × n rectangular Vandermonde matrix such that m ≥ n has rank n if and only if there are n of the x i that are distinct. A square Vandermonde matrix is invertible if and only if the x i are distinct. An explicit formula for the inverse is known (see below). [10] [3]
Download as PDF; Printable version; ... is the companion matrix of the polynomial), ... One can obtain explicit formulas for the above expressions in the form of ...
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