By Anthony Ralston

Notable textual content treats numerical research with mathematical rigor, yet particularly few theorems and proofs. orientated towards machine ideas of difficulties, it stresses blunders in equipment and computational potency. difficulties — a few strictly mathematical, others requiring a working laptop or computer — look on the finish of every bankruptcy

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Q☎ ❁✸ ❁✸✷❃❄✳ ✷✳✹✸✺✵ ✆☎✆✌✂✂✝ ✟ ✁ ✄☛✡ ✆ ➒ ✆✍☞✎✟ ✠✏✆ ✁ ✆ ✼✻ ✄ The field of values of an arbitrary matrix is a convex set which contains the convex hull of its spectrum. It is equal to the convex hull of the spectrum when the matrix is normal. ✂✁✞✝✆✁✯✮ ✙☞✟✡✞✆✩ ✏ ✎✴✏ ✛✴✔ ✩ ✛ ✎ ✞✣✏✷✌ ✟ ✝ A first result on Hermitian matrices is the following. , ✿ Let ✭ be an eigenvalue of Then ✭ ⑩ ✯ ⑩ ✰ ✑✁ . and ✽ an associated eigenvector or 2-norm unity. ❼✙✯ ⑩ ❺ ✰ ❼ ✯ ❺ ⑩ ✰ ❼ ✯ ⑩ ❺ ✰ ❼ ❺ ✾✽ ✽ ✽ ✾✽ ✾✽ ✽ ✭ which is the stated result.

Therefore, An equally important result is stated in the following theorem. ☎✁❃✆ ✴✝❥✆✰✞✔➒ ✸ ✼✻ ✄ The series ✁ ✁ ⑩ ✰ ✒ ❽✣ ⑩✰ ✻ ⑩✛ ✏ . Under this condition, ✦ ✚ . 4. In☞ ☞ ☎ ✪ deed, if the series converges, then . By the previous theorem, this implies that . To show that the converse is also true, use the equality ✩ ✛✧ ✣ ✯✏ ✦ ✛ ✏ ✦ ✯✏ ⑩ ✰ ➍➋ ⑩✓➋➍⑩ ➋ ❿✤❿✢❿ ➋ ⑩ ✰ ✁✣ ✠✯ ⑩ ✰ ✒✸✛ ❽ ✣ ⑩ ✩ ⑩ ✰ ✯ ⑩ ✰ ❼ ➋➍⑩✓➋➍⑩ ➋ ❿✤❿✢❿ ➋ ⑩ ✛ ❿ ⑩ ❼ and exploit the fact that since ✏ ✦ ✯✏ ✦ ✚ ✧ ✏ ✦ , then is nonsingular, and therefore, ✏ This shows that the series converges since the left-hand side will converge to In addition, it also shows the second part of the theorem.

Assume that The proof is by induction. 35) is true for ✎ . 6. 37) show that proof. ✧ ❏ ✆✞✝ ✂✂✜ ✟ , which completes the induction A theorem which has important consequences on the analysis of iterative methods will now be stated. 6 ✁ ✂ ✁ for any matrix norm. 6 ✁ ✁ ✂ ✁ ✂ ✁ ✰ ☎✁❃✆ ✴✝❥✆✰✞✔➒ ❚➒ ✄ Let ➌ be✣ a nonnegative matrix. Then ✁ ✯ ➌ ❾✒ ❽ if and only if ➌ ✰ is nonsingular and ✯ ➌ ✰ is nonnegative. ✄ ✆✞ ✞ ✡ ✝ ❼ ➌ ✁ ✯➌ ✰ ✒ ❽ ➊ ❼➊ ➌ ✆✞✝ ❏ ✄ ✟ ➊ ✣ ❼ ✯ ➌ ✰ ✣ ❼ ✚❷ ✜ ➌ ❷ ❿ ➌ ✜ ➌ ➊ ✁✠✯ ➌ ✰ ➌ ❼ ✁ ✯➌ ✰ ➊ ✣ ❼ ❽ ❽✁✠✯ ➌ ❿ ✰ ❽ ✁ ✯➌ ✰ ✏ ➊ ✣ ➌ which completes the proof.