A Symmetric Numerical Range for Matrices - Semantic Scholar
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A Symmetric Numerical Range for Matrices - Semantic Scholar
A Symmetric Numerical Range for Matrices B. David Saunders and Hans Schneider* Received October 2, 1975
Summary . For each norm v on 1) +a11.. sJ. e- i (Of-,pl)) ,
iEN\U}
since, for jEE, we have si=1 and (Ji=¢i' Hence I~i-aiil ~t L (jaiil+laiii)· iEN\{i}
Since Lri=1,itfollowsthatt(y*ax+x*aY)EG(a).
0
iEE
(2.7)
Example. Let
a=[~ ~] and let v be the 11 norm. Then
V(a) C Z. (a) C y. (a)
I l Y.D
(a) C G (a)
where all the containments are strict. For, V(a) = [0,5], Z.(a) = [-1,6], G(a)=conv(G1 (a), G2 (a)), where G1 (a) and G2 (a) are the circles of radius 2 with center 1 and 4 respectively. By NirschlSchneider [11], y'(a) 2G i (a), i=1, 2 and y'(a) is non-convex since the segment [1 + 2i, 4+ 2iJ intersects y. (a) in the points {1 + 2i, 4+ 2i} only. We also have y'(a) = y'D(a). The last remark is a consequence of the next result whose proof is easy. (2.8) Lemma. Let v be a norm on
