Joint ranges of Hermitian matrices and simultaneous diagonalization

Binding, Paul; Li, Chi-Kwong

doi:10.1016/0024-3795(91)90361-y

Cited by 39 publications

(24 citation statements)

References 11 publications

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“…Proof. The assertion on W (F, G, K, H) is well known; see [1]. The assertion on S + H (A) follows readily from Proposition 2.4.…”

Section: And Therefore the Cross Section Of Smentioning

confidence: 84%

Shells of matrices in indefinite inner product spaces

Bolotnikov¹,

Li²,

Meade³

et al. 2002

ELA

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Abstract. The notion of the shell of a Hilbert space operator, which is a useful generalization (proposed by Wielandt) of the numerical range, is extended to operators in spaces with an indefinite inner product. For the most part, finite dimensional spaces are considered. Geometric properties of shells (convexity, boundedness, being a subset of a line, etc.) are described, as well as shells of operators in two dimensional indefinite inner product spaces. For normal operators, it is conjectured that the shell is convex and its closure is polyhedral; the conjecture is proved for indefinite inner product spaces of dimension at most three, and for finite dimensional inner product spaces with one positive eigenvalue.

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“…Proof. The assertion on W (F, G, K, H) is well known; see [1]. The assertion on S + H (A) follows readily from Proposition 2.4.…”

Section: And Therefore the Cross Section Of Smentioning

confidence: 84%

Shells of matrices in indefinite inner product spaces

Bolotnikov¹,

Li²,

Meade³

et al. 2002

ELA

Self Cite

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“…JC 3 _ 7 -(0,'JC3->(0}-It follows that the points J{ 1 (JC,(O) and/,"' (x 2 (t)) in §(X(r)) are stationary points of the numerical range map n P(t)Ah of P(t) A|x (r) , and, by Remark 2.7, that*' 1 ({M0})Un;'({M 2 (0}) = E 0 (/'(r)^^,))-Hence, 7,"' is the line segment joining the points n x (f) and fi 2 …”

Section: Tj-\ U U)) S(x(t)) -{J^ixju))} 1 N X(t) Of S(x(r)) At Jf L (mentioning

confidence: 91%

“…Here, A t and A 2 are the uniquely determined Hermitian operators in jSf (C) such that A = Ai + iA 2 . By identifying a point (p,q) e R" x R" = R 2n with the vector p + iq e O , the map n A can be viewed as the Rayleigh quotient of A restricted to S(C").…”

Section: ") -> R 2 Of a Which Is Given By N A (Pq) = {(A I {P + Iq)mentioning

confidence: 99%

“…Following the approach in [8], we split the domain §(R 2 ") of n A into the three subsets E 0 (A), £,04) and T, 2 (A). Here, E y -( A) is the set of all points (p,q) e §(R 2 ") such that the derivative n' A (p, q) of n A at (p, q), which is a linear map from the (2« -1)-dimensional tangent space to §(R 2 ") at (p, q) into R [3] On the numerical range map 269…”

Section: ") -> R 2 Of a Which Is Given By N A (Pq) = {(A I {P + Iq)mentioning

confidence: 99%

“…s . x through the point /x 0 with coordinates (c, s, k) such that A. is an eigenvalue of the operator c 4 , +sA 2 . We pick a line l Co , So The line / 1O JLO intersects the boundary of /,"' (VK(/4)) in two points fi t and /z 2 -For j = 1, 2, take (pj, qj) e «^'({/Li y }) and put Xj = p 7 + iqj.…”

Section: ") -> R 2 Of a Which Is Given By N A (Pq) = {(A I {P + Iq)mentioning

confidence: 99%

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On the numerical range map

Joswig

Straub

1998

J Aust Math Soc A

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Let A e J2f(C) and A t , A 2 be the unique Hermitian operators such that A = A ] + iA 2 . The paper is concerned with the differential structure of the numerical range map n A : x i->• ({A\X, x). {A 2 x, v>) and its connection with certain natural subsets of the numerical range W( A) of A. We completely characterize the various sets of critical and regular points of the map n A as well as their respective images within W(A). In particular, we show that the plane algebraic curves introduced by R. Kippenhahn appear naturally in this context. They basically coincide with the image of the critical points of n A .

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