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Topics in Matrix Analysis by Charles R. Johnson, Roger A. Horn

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1.6    Geometry

In Section (1.3) we saw that if A ϵ M2, then the field of values F(A) is a (possibly degenerate) ellipse with interior and that the ellipse may be determined in several ways from parameters associated with the entries of the matrix A.

Exercise. If image, show that F(A) is: (a) a point if and only if λ2 = λ1 and β = 0; (b) a line segment joining λ1 and λ2 if and only if β = 0; (c) a circular disc of radius image if and only if λ2 = λ1; or (d) an ellipse (with interior) with foci at λ1 and λ2 otherwise, and λ1 and λ2 are interior points of the ...

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