v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
554 APPENDIX A. LINEAR ALGEBRA Now let A,B ∈ S M have diagonalizations A=QΛQ T and B =UΥU T with λ(A)=δ(Λ) and λ(B)=δ(Υ) arranged in nonincreasing order. Then where S = QU T . [344,7.5] A ≽ B ⇔ λ(A−B) ≽ 0 (1400) A ≽ B ⇒ λ(A) ≽ λ(B) (1401) A ≽ B λ(A) ≽ λ(B) (1402) S T AS ≽ B ⇐ λ(A) ≽ λ(B) (1403) A.3.1.0.2 Theorem. (Weyl) Eigenvalues of sum. [176,4.3.1] For A,B∈ R M×M , place the eigenvalues of each symmetrized matrix into the respective vectors λ ( 1 (A 2 +AT ) ) , λ ( 1 (B 2 +BT ) ) ∈ R M in nonincreasing order so λ ( 1 (A 2 +AT ) ) holds the largest eigenvalue of symmetrized A while 1 λ ( 1 (B 2 +BT ) ) holds the largest eigenvalue of symmetrized B , and so on. 1 Then, for any k ∈{1... M } λ ( A +A T) k + λ( B +B T) M ≤ λ( (A +A T ) + (B +B T ) ) k ≤ λ( A +A T) k + λ( B +B T) 1 ⋄ (1404) Weyl’s theorem establishes positive semidefiniteness of a sum of positive semidefinite matrices. Because S M + is a convex cone (2.9.0.0.1), then by (157) A,B ≽ 0 ⇒ ζA + ξB ≽ 0 for all ζ,ξ ≥ 0 (1405) A.3.1.0.3 Corollary. Eigenvalues of sum and difference. [176,4.3] For A∈ S M and B ∈ S M + , place the eigenvalues of each matrix into the respective vectors λ(A), λ(B)∈ R M in nonincreasing order so λ(A) 1 holds the largest eigenvalue of A while λ(B) 1 holds the largest eigenvalue of B , and so on. Then, for any k ∈{1... M} λ(A − B) k ≤ λ(A) k ≤ λ(A +B) k (1406) ⋄ When B is rank-one positive semidefinite, the eigenvalues interlace; id est, for B = qq T λ(A) k−1 ≤ λ(A − qq T ) k ≤ λ(A) k ≤ λ(A + qq T ) k ≤ λ(A) k+1 (1407) ⋄
A.3. PROPER STATEMENTS 555 A.3.1.0.4 Theorem. Positive (semi)definite principal submatrices. A.10 A∈ S M is positive semidefinite if and only if all M principal submatrices of dimension M−1 are positive semidefinite and detA is nonnegative. A ∈ S M is positive definite if and only if any one principal submatrix of dimension M −1 is positive definite and detA is positive. ⋄ If any one principal submatrix of dimension M−1 is not positive definite, conversely, then A can neither be. Regardless of symmetry, if A∈ R M×M is positive (semi)definite, then the determinant of each and every principal submatrix is (nonnegative) positive. [235,1.3.1] A.3.1.0.5 [176, p.399] Corollary. Positive (semi)definite symmetric products. If A∈ S M is positive definite and any particular dimensionally compatible matrix Z has no nullspace, then Z T AZ is positive definite. If matrix A∈ S M is positive (semi)definite then, for any matrix Z of compatible dimension, Z T AZ is positive semidefinite. A∈ S M is positive (semi)definite if and only if there exists a nonsingular Z such that Z T AZ is positive (semi)definite. If A∈ S M is positive semidefinite and singular it remains possible, for some skinny Z ∈ R M×N with N
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A.3. PROPER STATEMENTS 555<br />
A.3.1.0.4 Theorem. Positive (semi)definite principal submatrices. A.10<br />
A∈ S M is positive semidefinite if and only if all M principal submatrices<br />
of dimension M−1 are positive semidefinite and detA is nonnegative.<br />
A ∈ S M is positive definite if and only if any one principal submatrix<br />
of dimension M −1 is positive definite and detA is positive. ⋄<br />
If any one principal submatrix of dimension M−1 is not positive definite,<br />
conversely, then A can neither be. Regardless of symmetry, if A∈ R M×M is<br />
positive (semi)definite, then the determinant of each and every principal<br />
submatrix is (nonnegative) positive. [235,1.3.1]<br />
A.3.1.0.5<br />
[176, p.399]<br />
Corollary. Positive (semi)definite symmetric products.<br />
If A∈ S M is positive definite and any particular dimensionally<br />
compatible matrix Z has no nullspace, then Z T AZ is positive definite.<br />
If matrix A∈ S M is positive (semi)definite then, for any matrix Z of<br />
compatible dimension, Z T AZ is positive semidefinite.<br />
A∈ S M is positive (semi)definite if and only if there exists a nonsingular<br />
Z such that Z T AZ is positive (semi)definite.<br />
If A∈ S M is positive semidefinite and singular it remains possible, for<br />
some skinny Z ∈ R M×N with N
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