On eigenvalues, eigenvectors and singular values in robust stability analysis†
نویسندگان
چکیده
منابع مشابه
On computing accurate singular values and eigenvalues . . .
[15] D. O'Leary and G. W. Stewart. Computing the eigenvalues and eigenvectors of symmetric arrowhead matrices. [17] A. Sameh and D. Kuck. A parallel QR algorithm for symmetric tridiagonal matrices. [21] Zhonggang Zeng. The acyclic eigenproblem can be reduced to the arrowhead one. [22] Hongyuan Zha. A two-way chasing scheme for reducing a symmetric arrowhead matrix to tridiagonal form. Scientic ...
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for some nonzero vector x called an eigenvector. This is equivalent to writing (λI − A)x = 0 so, since x 6= 0, A− λI must be singular, and hence det(λI − A) = 0. From the (complicated!) definition of determinant, it follows that det(λI−A) is a polynomial in the variable λ with degree n, and this is called the characteristic polynomial. By the fundamental theorem of algebra (a nontrivial result)...
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Exercise 4. Let λ be an eigenvalue of A and let Eλ(A) = {x ∈ C|Ax = λx} denote the set of all eigenvectors of A associated with λ (including the zero vector, which is not really considered an eigenvector). Show that this set is a (nontrivial) subspace of C. Definition 5. Given A ∈ Cm×m, the function pm(λ) = det(λI − A) is a polynomial of degree at most m. This polynomial is called the character...
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Chi-Kwong Li Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795, USA E-mail: [email protected] We briefly describe some recent results on inequalities relating the eigenvalues of the sum of Hermitian or real matrices, and how to use these them inequalities relating the eigenvalues and singular values of a matrix and its submatrices. These results are joint ...
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This chapter is about eigenvalues and singular values of matrices. Computational algorithms and sensitivity to perturbations are both discussed. An eigenvalue and eigenvector of a square matrix A are a scalar λ and a nonzero vector x so that Ax = λx. A singular value and pair of singular vectors of a square or rectangular matrix A are a nonnegative scalar σ and two nonzero vectors u and v so th...
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ژورنال
عنوان ژورنال: International Journal of Control
سال: 1984
ISSN: 0020-7179,1366-5820
DOI: 10.1080/00207178408933275