Singular Values and Singular Value Inequalities 17-7 Examples: 1.The matrix A in Example 1 of Section 17.1 has singular values20,128,and 4.So IIAll2 =20,IlAllF =24,IlAllr =44; 1AlK.1=20,1Ax2=32,IAK.3=40,1AK4=44 1Als.1=44,1As2=√624，IAls3=√/10304=21.7605,1A5.x=20. 17.4 Inequalities Definitions: Pinching is defined recursively.If A21 A22 of B is Facts: Thbe found dad eor eamle Chap.5 ules anothe 1.(Submatrices)Take AC and let B denote A with one of its rows or columns deleted.Then 之A公om dod.Then o42(A)≤o(B)≤0(A),i=1,...,9-2 Thei+2 ca B be an (m-k)x(n-1)submatrix of A.Then O+k+H(A)≤a(B)≤o(A,i=1,,q-(k+I). 4.Take AC and let B be A with some of its rows and/or columns set to zero.Then(B)s A) 1 5.Let B bea qualitiesΠ1(B)≤o(A)and (B)< not n arily true fork>1.(Example 1) 6.(Singular values ofA+B)Let A.BC (a)sv(A+B)sv(A)+sv(B),or equivalently ∑o(A+B)≤∑o,(A)+∑0(B,i=1,,9 i=1 (b)Ifi+j-1s q andi,jEN,then (A+B)s(A)+aj(B). Singular Values and Singular Value Inequalities 17-7 Examples: 1. The matrix A in Example 1 of Section 17.1 has singular values 20, 12, 8, and 4. So A2 = 20, AF = √624, Atr = 44; AK,1 = 20, AK,2 = 32, AK,3 = 40, AK,4 = 44; AS,1 = 44, AS,2 = √624, AS,3 = √3 10304 = 21.7605, AS,∞ = 20. 17.4 Inequalities Throughout this section, q = min{m, n} and if A ∈ Cm×n has real eigenvalues, then they are ordered λ1(A) ≥···≥ λn(A). Definitions: Pinching is defined recursively. If A =
A11 A12 A21 A22 ∈ Cm×n, B =
A11 0 0 A22 ∈ Cm×n, then B is a pinching of A. (Note that we do not require the Aii to be square.) Furthermore, any pinching of B is a pinching of A. For positive α, β, define the measure of relative separation χ(α, β) = |√α/β − √β/α|. Facts: The following facts can be found in standard references, for example [HJ91, Chap. 3], unless another reference is given. 1. (Submatrices) Take A ∈ Cm×n and let B denote A with one of its rows or columns deleted. Then σi+1(A) ≤ σi(B) ≤ σi(A), i = 1, ... , q − 1. 2. Take A ∈ Cm×n and let B be A with a row and a column deleted. Then σi+2(A) ≤ σi(B) ≤ σi(A), i = 1, ... , q − 2. The i + 2 cannot be replaced by i + 1. (Example 2) 3. Take A ∈ Cm×n and let B be an (m − k) × (n − l) submatrix of A. Then σi+k+l(A) ≤ σi(B) ≤ σi(A), i = 1, ... , q − (k + l). 4. Take A ∈ Cm×n and let B be A with some of its rows and/or columns set to zero. Then σi(B) ≤ σi(A), i = 1, ... , q. 5. Let B be a pinching of A. Then sv(B) w sv(A). The inequalities k i=1 σi(B) ≤ k i=1 σi(A) and σk (B) ≤ σk (A) are not necessarily true for k > 1. (Example 1) 6. (Singular values of A + B) Let A, B ∈ Cm×n. (a) sv(A + B) w sv(A) + sv(B), or equivalently k i=1 σi(A + B) ≤ k i=1 σi(A) + k i=1 σi(B), i = 1, ... , q. (b) If i + j − 1 ≤ q and i, j ∈ N, then σi+ j−1(A + B) ≤ σi(A) + σj(B).