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The Characteristic Polynomial
"... Consider the eigenvalue problem for an n × n matrix A, Av = λv, v = 0. (1) The solution to this problem consists of identifying all possible values of λ (called the eigenvalues), and the corresponding nonzero vectors v (called the eigenvectors) that satisfy eq. (1). Noting that Iv = v, one can rew ..."
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nontrivial solutions to eq. (2), one must demand that A −λI is not invertible, or equivalently, p(λ) ≡ det(A − λI) = 0. (3) Eq. (3) is called the characteristic equation. Evaluating the determinant yields an nth order polynomial in λ, called the characteristic polynomial, which we have denoted
CHARACTERISTIC POLYNOMIALS
"... Introduction. Let F be a field and let V be a finite dimensional vector space over F which is also a module over the ring F[a]. Here a may lie in any extension ring of F. We do not assume, as yet, that V is a faithful module, so that a need not be a linear transformation on V. It is known that by me ..."
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Introduction. Let F be a field and let V be a finite dimensional vector space over F which is also a module over the ring F[a]. Here a may lie in any extension ring of F. We do not assume, as yet, that V is a faithful module, so that a need not be a linear transformation on V. It is known that by means of a decomposition of V into cyclic F[a]modules we may obtain a definition
The characteristic polynomial of a multiarrangement
, 2006
"... Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteris ..."
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Cited by 20 (11 self)
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Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes
Characteristic polynomials and pseudospectra
, 2003
"... In this paper, we study the εlemniscate of the characteristic polynomial in relation to the pseudospectrum of the associated matrix. It is natural to investigate this question because these two sets can be seen as generalizations of eigenvalues. The question of numerical determination of the εlemn ..."
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In this paper, we study the εlemniscate of the characteristic polynomial in relation to the pseudospectrum of the associated matrix. It is natural to investigate this question because these two sets can be seen as generalizations of eigenvalues. The question of numerical determination of the ε
Characteristic polynomials of random matrices
 Communications in Mathematical Physics
"... We have discussed earlier the correlation functions of the random variables det(λ−X) in which X is a random matrix. In particular the moments of the distribution of these random variables are universal functions, when measured in the appropriate units of the level spacing. When the λ’s, instead of b ..."
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Cited by 70 (0 self)
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and one can tune the spectrum of this source matrix to new critical points. Again there are remarkably simple formulae for arbitrary source matrices, which allow us to compute the moments of the characteristic polynomials in these cases as well.
Properties and Application of Characteristic Polynomial
"... Abstract. Through the introduction to the properties of characteristic polynomial, the order reduction theorem of the characteristic polynomial as well as its application in high order matrix is studied, and also a simplified method (characteristic polynomial method), which is used to solve the part ..."
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Abstract. Through the introduction to the properties of characteristic polynomial, the order reduction theorem of the characteristic polynomial as well as its application in high order matrix is studied, and also a simplified method (characteristic polynomial method), which is used to solve
CHARACTERISTIC POLYNOMIALS OF SUPERTROPICAL MATRICES
"... Abstract. Supertropical matrix theory was investigated in [6], whose terminology we follow. In this work we investigate eigenvalues, characteristic polynomials and coefficients of characteristic polynomials of supertropical matrices and their powers, and obtain the analog to the basic property of ma ..."
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Abstract. Supertropical matrix theory was investigated in [6], whose terminology we follow. In this work we investigate eigenvalues, characteristic polynomials and coefficients of characteristic polynomials of supertropical matrices and their powers, and obtain the analog to the basic property
Efficient computation of the characteristic polynomial
 Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation
, 2005
"... We deal with the computation of the characteristic polynomial of dense matrices over word size finite fields and over the integers. We first present two algorithms for finite fields: one is based on Krylov iterates and Gaussian elimination. We compare it to an improvement of the second algorithm of ..."
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Cited by 18 (13 self)
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We deal with the computation of the characteristic polynomial of dense matrices over word size finite fields and over the integers. We first present two algorithms for finite fields: one is based on Krylov iterates and Gaussian elimination. We compare it to an improvement of the second algorithm
Results 1  10
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