**Unformatted text preview: **AU = λ
0 x∗ AVQ
Q∗ V∗ AVQ λ
0 = x∗ AVQ
˜
T =T is upper triangular. Since similar matrices have the same eigenvalues, and since
the eigenvalues of a triangular matrix are its diagonal entries (Exercise 7.1.3),
the diagonal entries of T must be the eigenvalues of A. D
E Example 7.2.2
71 The Cayley–Hamilton theorem asserts that every square matrix satisﬁes
its own characteristic equation p(λ) = 0. That is, p(A) = 0. T
H Problem: Show how the Cayley–Hamilton theorem follows from Schur’s triangularization theorem.
Solution: Schur’s theorem insures the existence of a unitary U such that
U∗ AU = T is triangular, and the development allows for the eigenvalues A to
appear in any given order on the diagonal of T. So, if σ (A) = {λ1 , λ2 , . . . , λk }
with λi repeated ai times, then there is a unitary U such that
⎞
⎞
⎛
⎛
···
···
T1
λi
T2 · · ·
λi · · ·
⎟
⎟
⎜
⎜
U∗ AU = T = ⎜
. ⎟, where Ti = ⎜
. ⎟.
..
..
⎝
⎝
.⎠
.⎠
.
..
. Y
P IG
R Tk λi ai ×ai Consequently, (Ti − λi I) = 0, so (T − λi I) has the form
⎛
⎞
···
···
.⎟
.
..
⎜
.⎟
..
.
.
⎜
⎜
⎟
ai
(T − λi I) = ⎜
0 ···
⎟ ←− ith row of blocks.
⎜
.⎟
..
⎝
. .⎠
.
ai 71 ai O
C William Rowan Hamilton (1805–1865), an Irish mathematical astronomer, established this
a + bi c + di
result in 1853 for his quaternions, matrices of the form
that resulted
−c + di a − bi
from his attempt to generalize complex numbers. In 1858 Arthur Cayley (p. 80) enunciated
the general result, but his argument was simply to make direct computations for 2 × 2 and
3 × 3 matrices. Cayley apparently didn’t appreciate the subtleties of the result because he
stated that a formal proof “was not necessary.” Hamilton’s quaternions took shape in his mind
while walking with his wife along the Royal Canal in Dublin, and he was so inspired that he
stopped to carve his idea in the stone of the Brougham Bridge. He believed quaternions would
revolutionize mathematical physics, and he spent the rest of his life working on them. But...

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