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**Unformatted text preview: **∗ BV = D
y∗ y
α , y = U∗ c. where For x ∈ F = span {e1 , e2 , . . . , ei } ⊂ C n+1×1 with x 2 = 1, n x = (x1 , . . . , xi , 0, . . . , 0)T and ˜
x∗ Bx = n λj |xj |2 ≥ λi
j =1 |xj |2 = λi ,
j =1 ˜
so applying the max-min part of the Courant–Fisher theorem to B yields
βi = max dim V =i Copyright c 2000 SIAM ˜
˜
min x∗ Bx ≥ min x∗ Bx ≥ λi .
x∈V
x 2 =1 x∈F
x 2 =1 Buy online from SIAM
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7.5 Normal Matrices
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For x ∈ T = span{ei−1 , ei , . . . , en } ⊂ C n+1×1 with x 2 553 = 1, n n ˜
x = (0, . . . , 0, xi−1 , . . . , xn , 0)T and x∗ Bx = λj |xj |2 ≤ λi−1 j =i−1 |xj |2 = λi−1 ,
j =i so the min-max part of the Courant–Fisher theorem produces It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] βi = min dim V =n−i+2 ˜
˜
max x∗ Bx ≤ max x∗ Bx ≤ λi−1 .
x∈V
x 2 =1 x∈F
x 2 =1 D
E Note: If A is any n × n principal submatrix of B, then (7.5.8) still holds
because each principal submatrix can be brought to the upper-left-hand corner
by a similarity transformation PT BP, where P is a permutation matrix. In
other words,
• the eigenvalues of an n + 1 × n + 1 hermitian matrix are interlaced with the
eigenvalues of each of its n × n principal submatrices.
m×n m×n T
H ∗ For A ∈ C
(or
), the products A A and AA∗ (or AT A and
AA ) are hermitian (or real symmetric), so they are diagonalizable by a unitary (or orthogonal) similarity transformation, and their eigenvalues are necessarily real. But in addition to being real, the eigenvalues of these matrices
are always nonnegative. For example, if (λ, x) is an eigenpair of A∗ A, then
2
2
λ = x∗ A∗ Ax/x∗ x = Ax 2 / x 2 ≥ 0, and similarly for the other products. In
fact, these λ ’s are the squares of the singular values for A developed in §5.12
(p. 411) because if
Dr×r 0
A=U
V∗
0
0 m×n IG
R T Y
P O
C is a singular value decomposition, where D = diag (σ1 ,...

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