Some angularity and inertia theorems related to normal matrices
نویسندگان
چکیده
منابع مشابه
Some Theorems on the Inertia of General Matrices
1.1. Much is known about the distribution of the roots of algebraic equations in half-planes. (Cf. the corresponding parts in the survey [l] by Marden.) In the case of matrix equations, however, there appears to be only one known general result concerning the location of the eigenvalues of a matrix in the left half-plane. This theorem is generally known as Lyapunov’s theorem: (L,) Let A be an n...
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1. The inertia of a square matrix A with complex elements is defined to be the integer triple In A = (ir(A), V(A), 8(4)), where ir(A) {v(A)} equals the number of eigenvalues in the open right {left} half plane, and 8(A) equals the number of eigenvalues on the imaginary axis. The best known classical inertia theorem is that of Sylvester : If P > 0 (positive definite) and H is Hermitian, then In ...
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We present a new proof and extension of the classical Sylvester Inertia Theorem to a pair of non-Hermitian matrices which satisfies the property that any real linear combination of the pair has only real eigenvalues. In the proof, we embed the given problem in a one-parameter family of related problems and examine the eigencurves of the family. The proof requires only elementary matrix theory a...
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The study of sum-product phenomena and product phenomena is an emerging research direction in combinatorial number theory that has already produced several striking results. Many related problems are not yet fully understood, or are far from being resolved. In what follows we propose several questions where progress can be expected and should lead to advances in this general area. For two finit...
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ژورنال
عنوان ژورنال: Linear Algebra and its Applications
سال: 1981
ISSN: 0024-3795
DOI: 10.1016/0024-3795(81)90140-3