By Dennis G. Zill
Written for junior-level undergraduate scholars which are majoring in math, physics, machine technological know-how, and electric engineering.
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Prove your assertion. (c) What can be said about the entries on the main diagonal of a skewHermitian matrix? Prove your assertion. (d) Prove that the eigenvalues of a Hermitian matrix are real. (e) Prove that the eigenvalues of a skew-Hermitian matrix are either pure imaginary or zero. (f ) Prove that the eigenvalues of unitary matrix are unimodular; that is, |λ| = 1. Describe where these eigenvalues are located in the complex plane. (g) Prove that the modulus of a unitary matrix is one, that is, |det A| = 1.
All complex numbers z = x + iy for which x > 1 are in this set. 05. 17. On the other hand, the set S of points in the complex plane deﬁned by Re(z) ≥ 1 is not open because every neighborhood of a point lying on the line x = 1 must contain points in S and points not in S. 18. 19 illustrates some additional open sets. 5 Sets of Points in the Complex Plane If every neighborhood of a point z0 of a set S contains at least one point of S and at least one point not in S, then z0 is said to be a boundary point of S.
In Problems 1–14, use (4) to compute all roots. Give the principal nth root in each case. Sketch the roots w0 , w1 , . . , wn−1 on an appropriate circle centered at the origin. 1. (8)1/3 2. (−1)1/4 3. (−9)1/2 4. (−125)1/3 5. (i)1/2 6. (−i)1/3 7. (−1 + i)1/3 √ 9. (−1 + 3i)1/2 11. (3 + 4i)1/2 13. 16i 1+i 8. (1 + i)1/5 √ 10. (−1 − 3i)1/4 12. (5 + 12i)1/2 1/8 14. 1+i √ 3+i 1/6 15. (a) Verify that (4 + 3i)2 = 7 + 24i. (b) Use part (a) to ﬁnd the two values of (7 + 24i)1/2 . 4 Powers and Roots 27 16.
A First Course in Complex Analysis with Applications by Dennis G. Zill