j 0 i n where n z0 ij converges 1

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Unformatted text preview: didn’t teach or hold a o university appointment, but his research activities led to his election to the Berlin Academy in 1860. He declined the offer of the mathematics chair in G¨ttingen in 1868, but he eventually o accepted the chair in Berlin that was vacated upon Kummer’s retirement in 1883. Kronecker held the unconventional view that mathematics should be reduced to arguments that involve only integers and a finite number of steps, and he questioned the validity of nonconstructive existence proofs, so he didn’t like the use of irrational or transcendental numbers. Kronecker became famous for saying that “God created the integers, all else is the work of man.” Kronecker’s significant influence led to animosity with people of differing philosophies such as Georg Cantor (1845–1918), whose publications Kronecker tried to block. Kronecker’s small physical size was another sensitive issue. After Hermann Schwarz (p. 271), who was Kummer’s son-in-law and a student of Weierstrass (p. 589), tried to make a joke involving Weierstrass’s large physique by stating that “he who does not honor the Smaller, is not worthy of the Greater,” Kronecker had no further dealings with Schwarz. O C Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 598 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 or direct product ) of Am×n and Bp×q is the mp × nq matrix ⎛ ⎞ a1n B a2n B ⎟ . ⎟. .⎠ . (a) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ (b) a12 B a22 B . . . ··· ··· .. . am1 B It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu a11 B ⎜ a21 B A⊗B=⎜ . ⎝. . am2 B · · · amn B Assuming conformability, establish the following properties. A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C. A ⊗ (B + C) = (A ⊗ B) + (A ⊗ C). (A + B) ⊗ C = (A ⊗ C) + (B ⊗ C). (A1 ⊗ B1 )(A2 ⊗ B2 ) · · · (Ak ⊗ Bk ) = (A1 · · · Ak ) ⊗ (B1 · ·...
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