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Unformatted text preview: λ1 ∗ · · · ∗
. = 0
,
.
.
0 B 2 where B ∈ Mn−1 . Now, the characteristic polynomial of A factors as:
PA ( t ) = ( t − λ1 ) PB ( t ) ,
where PB (t) is the characteristic polynomial of B , which means {λ2 , λ3 , ..., λn } must be eigenvalues for B . Now we are in a position to invoke the induction assumption: there exists
U ∈ Mn−1 such that U (U )∗ = I and (U )∗ BU = T , where T ∈ Mn−1 is upper triangular with λ2 , λ3 , ..., λn on the diagonal . Now deﬁne U2 as: 1 0···0
. U2 = 0
,
.
.
0 U and deﬁne U as U = U1 U2 , which is unitary since products of unitary matriecs are unitary. Then:
∗∗
∗
∗
U ∗ AU = (U1 U2 )∗ A(U1 U2 ) = U2 U1 AU1 U2 = U2 (U1 AU1 )U2 λ1
∗··...
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 Fall '12
 BernhardBodmann
 Matrices

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