It follows that, since the related linear equations are solved with the necessary precision, the eigenvalues of pejorative matrices can be calculated numerically arbitrarily and thus also the corresponding (generalized) eigenvectors. A pejorative matrix is a matrix that is not non-pejorative. 4] The characteristic matrix and minimal polynomial of a matrix nxn A are identical if and only if A has only one nontrivial similarity invariant. be a monic polynomial. We define the companion matrix of g(λ) as the matrix nxn C(g) elementary divisor of λI – D, where D is a diagonal matrix. Let D be a 1] Let f(λ) be the characteristic polynomial and m(λ) be the minimal polynomial of a matrix nxn A. Then Def. Invariants of similarity of a matrix. Let A be an nxn matrix whose elements are numbers of a number field F. The similarity invariants of the matrix A are the invariant factors of its characteristic matrix λI – A.

Since the characteristic matrix λI – A of A is nonsingular and n has invariant factors, the matrix A n has similarity invariants. The minimal polynomial and the minimum equation of a matrix. Each quadratic matrix satisfies its own characteristic equation (Hamilton–Cayley theorem), but there may be another equation of lower degree than the characteristic equation satisfying the matrix. The monic polynomial m(λ) of minimal degree, such that m(A) = 0 is called the minimum polynomial of A and m(A) = 0 is called the minimal equation of the matrix A. The original meaning derives from the Latin derogare, „to remove, distract, reduce“, see EtymOnline. The association with insult dates back to the late 16th century. For matrices, Sylvester`s term was introduced in the early 1880s in his matrix revision of Hamilton`s theory of quaternions, see The Emergence of the American Mathematical Research Community, 1876-1900, by Parshall and Rowe, p. 136. „Reduction“ probably refers to the degree of the characteristic polynomial that is reduced in the minimal polynomial.

On the other hand, Sylvester`s other proposal to rename the characteristic to a „latent“ polynomial did not prevail. ● If A with the nontrivial similarity invariant fn(λ) = (λ – a)n is not pejorative, then a condition of similarity to a diagonal matrix. An n-square matrix A over a field F looks like a diagonal matrix if and only if λ I − A has linear elementary divisors in F[λ]. 3] The characteristic matrix λ I – A of a matrix nxn A has the nmonic polynomials p1(λ), p2(λ), . , pn(λ) as 1., 2. ,. , n the diagonal elements of its Smith normal form (i.e. these n monic factors represent its n invariant factors arranged in order). Then the minimal polynomial of A is pn(λ).

● An n-square matrix A is nonpejorative if and only if A has a nontrivial similarity invariant. Elementary divisors of a matrix λ. Invariant factors in the Smith normal form ● Let g1(λ), g2(λ), . , gm(λ) are different, monic and irreducible polynomials in F[λ] and let Aj be a non-pejorative matrix, such that the minimal polynomial and the minimal equation of a matrix A. This monic Def. The characteristic matrix of a matrix. Let A be an nxn matrix whose elements are numbers of a number field F. The characteristic matrix of matrix A is the λ-matrix λI – A.

Properties of the characteristic matrix λI – A of a matrix A. If A is a matrix nxn over a field F, then its characteristic matrix λ I – A has the following special properties: 7] The characteristic matrix of an n-square matrix A has unique linear elementary divisors if and only if m(λ), the minimal polynomial of A, has only different linear factors. This companion matrix C(g) of the polynomial g(λ) has g(λ) both as a characteristic polynomial and as a minimal polynomial. This means that if a nonpejorative matrix A has the polynomial g(λ) as a single nontrivial similarity invariant, it is similar to the companion matrix C(g) of g(λ). In fact, C(g) of g(λ) will represent a canonical form for such a matrix. Characteristic curve of a matrix A. The matrix λI – A. My mother tongue is German and I know of only one place in German literature where non-pejorative matrices are given a special name. It is in E. Brieskorn „Linear Algebra and Analytic Geometry II“, where they are called „regular“, that the author immediately distinguishes from another use of this word as invertible. A quadratic matrix $A$ in which the characteristic polynomial and the minimal polynomial coincide (up to a factor of $pm1$). Equivalently, for each of its unique eigenvalues $lambda$ in the Jordan normal form for $A$, there is only one Jordan block with this eigenvalue $lambda$; This, in turn, corresponds to any single eigenvalue that has only one independent eigenvector, i.e.

geometric multiplicity. A matrix A is said to be pejorative if more than one Jordan submatrix is assigned to an eigenvalue Î. In this thesis, we deal with the problem of eigenvalues of this type of matrices. 5] If A is any n-square matrix over fields F and f(λ) is any polynomial over F, then f(A) = 0 is if and only if the minimal polynomial divides m(λ) from A f(λ). I would like to know because I teach advanced students in a German high school who will consider a field of study in the field of STEM, linear algebra using examples of polynomial geometry. In this context, some polynomials produce double-eigenvalued matrices, which are either non-pejorative or diagonalizable and in which non-pejorative matrices are actually preferable to diagonalizable ones because of the respective map. Characteristic matrix, similarity invariants, minimal polynomial, accompanying matrix, non-pejorative matrix Similarity invariant of a matrix A. An invariant factor of λI – A.

Characteristic roots of a matrix A. The roots of the characteristic equation theorem. The characteristic polynomial of an n-square matrix A is the product of the factors invariant of λI − A (or equivalent to the similarity invariants of A). 2] If the characteristic roots of a matrix are all different, the minimal polynomial is equal to (identical to) the characteristic matrix. ● Let λ1, λ2, . , λl is the set of all the unique roots of the characteristic equation of a matrix A, and n1, n2, . , nl are the multiplicities with which they occur. Then the characteristic polynomial f(λ) of the matrix is A non-pejorative matrix $A$ is a matrix whose minimal polynomial $m(z)$ is equal to its characteristic polynomial $p(z)$, where we apply the convention $p(z) = det(zI-A)$, while a matrix is pejorative if they do not coincide. I`ve certainly never felt particularly offended when it comes to the $$I identity matrix, so I seriously wonder where this strange name comes from. Non-pejorative matrix.

An nxn matrix whose characteristic polynomial and minimal polynomial are identical. Invariant factors of a matrix λ The monic polynomials f1(λ), f2(λ), . , fr(λ) We will now show a matrix which, both characteristic polynomial and minimal polynomial, is the polynomial The elementary divisors of the characteristic matrix λI – D of D In order to make the rather complicated term more acceptable to the audience, it is always good to tell a story about it, even if it gets rather boring at the end. This result can be extended to the direct sum of n matrices. are its diagonal elements λ – a1, λ – a2 , . 6] The characteristic polynomial f(λ) of A is the product of the minimal polynomial of A and some monic factors of m(λ). ● If matrices B1 and B2 have minim1(λ) and m2(λ) polynomials, respectively, the minimal polynomial m(λ) of the direct sum D = diag(B1,B2) is the smallest common multiple of m1(λ) and m2(λ).