TY - JOUR
T1 - Taylor expansions of eigenvalues of perturbed matrices with applications to spectral radii of nonnegative matrices
AU - Haviv, Moshe
AU - Ritov, Ya'acov
AU - Rothblum, Uriel G.
PY - 1992/4/15
Y1 - 1992/4/15
N2 - Let A and B be two n×n complex matrices, and let λ be an eigenvalue of A. The purpose of this paper is to derive, under certain conditions, Taylor power series expansions of the form λ+σ∞k=1λkεk and σ∞k=0v{script}kεk, respectively, for eigenvalues and corresponding eigenvectors of the perturbed matrices A+εB for ε that has sufficiently small absolute value. Our results apply to the case where λ is a simple eigenvalue of A, e.g., when A is nonnegative and irreducible and λ is the spectral radius of A. In particular, if A+εB is nonnegative for sufficiently small nonnegative ε and A is irreducible, we obtain power series expansions for the spectral radii of the perturbed matrices A+εB and for corresponding eigenvectors. The coefficients of the expansions yield explicit expressions for the regular and mixed derivatives of the spectral radius and of a corresponding eigenvector of a nonnegative irreducible matrix when viewed as a function of the elements of the matrix. Our approach is constructive, and we present a recursive algorithm that will compute the coefficients of the above series.
AB - Let A and B be two n×n complex matrices, and let λ be an eigenvalue of A. The purpose of this paper is to derive, under certain conditions, Taylor power series expansions of the form λ+σ∞k=1λkεk and σ∞k=0v{script}kεk, respectively, for eigenvalues and corresponding eigenvectors of the perturbed matrices A+εB for ε that has sufficiently small absolute value. Our results apply to the case where λ is a simple eigenvalue of A, e.g., when A is nonnegative and irreducible and λ is the spectral radius of A. In particular, if A+εB is nonnegative for sufficiently small nonnegative ε and A is irreducible, we obtain power series expansions for the spectral radii of the perturbed matrices A+εB and for corresponding eigenvectors. The coefficients of the expansions yield explicit expressions for the regular and mixed derivatives of the spectral radius and of a corresponding eigenvector of a nonnegative irreducible matrix when viewed as a function of the elements of the matrix. Our approach is constructive, and we present a recursive algorithm that will compute the coefficients of the above series.
UR - http://www.scopus.com/inward/record.url?scp=0007347092&partnerID=8YFLogxK
U2 - 10.1016/0024-3795(92)90293-J
DO - 10.1016/0024-3795(92)90293-J
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AN - SCOPUS:0007347092
SN - 0024-3795
VL - 168
SP - 159
EP - 188
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
IS - C
ER -