Unformatted text preview: didn’t teach or hold a
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
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
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Eigenvalues and Eigenvectors
or direct product ) of Am×n and Bp×q is the mp × nq matrix
a2n B ⎟
(b) a12 B
. am1 B It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] a11 B
⎜ a21 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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