ON MATRIX D -STABILITY AND RELATED PROPERTIES Olga Kushel Shanghai - - PowerPoint PPT Presentation

ON MATRIX D-STABILITY AND RELATED PROPERTIES

Olga Kushel

Shanghai Jiao Tong University, China

June 1, 2015

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Outline

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Outline

1 D-stable matrices Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Outline

1 D-stable matrices 2 Permutations and nested sequences of principal submatrices Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

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1 D-stable matrices 2 Permutations and nested sequences of principal submatrices 3 Dθ-stability Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

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1 D-stable matrices 2 Permutations and nested sequences of principal submatrices 3 Dθ-stability 4 P- and Q-matrices: introduction Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

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1 D-stable matrices 2 Permutations and nested sequences of principal submatrices 3 Dθ-stability 4 P- and Q-matrices: introduction 5 Stability of P-matrices: open problems Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Outline

1 D-stable matrices 2 Permutations and nested sequences of principal submatrices 3 Dθ-stability 4 P- and Q-matrices: introduction 5 Stability of P-matrices: open problems 6 Stabilization by a diagonal matrix Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Outline

1 D-stable matrices 2 Permutations and nested sequences of principal submatrices 3 Dθ-stability 4 P- and Q-matrices: introduction 5 Stability of P-matrices: open problems 6 Stabilization by a diagonal matrix 7 Stability of P2-matrices Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Outline

1 D-stable matrices 2 Permutations and nested sequences of principal submatrices 3 Dθ-stability 4 P- and Q-matrices: introduction 5 Stability of P-matrices: open problems 6 Stabilization by a diagonal matrix 7 Stability of P2-matrices 8 D-stability and Dθ-stability of P2-matrices Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Outline

1 D-stable matrices 2 Permutations and nested sequences of principal submatrices 3 Dθ-stability 4 P- and Q-matrices: introduction 5 Stability of P-matrices: open problems 6 Stabilization by a diagonal matrix 7 Stability of P2-matrices 8 D-stability and Dθ-stability of P2-matrices 9 Rank one perturbations of singular M-matrices Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

D-stable matrices

Definition An n × n matrix A is called positive stable or just stable if all its eigenvalues have positive real parts.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

D-stable matrices

Definition An n × n matrix A is called positive stable or just stable if all its eigenvalues have positive real parts. Definition An n × n matrix A is called D-stable if DA is stable for any positive diagonal matrix D.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Permutations and nested sequences of principal submatrices

Definition Given a positive diagonal matrix D = diag{d11, . . . , dnn} and a permutation θ = (θ(1), . . . , θ(n)) of the set of indices [n], we call the matrix D ordered with respect to θ or θ-ordered if it satisfies the inequalities dθ(i)θ(i) ≥ dθ(i+1)θ(i+1), i = 1, . . . , n − 1. We call the matrix D strictly θ-ordered if dθ(i)θ(i) > dθ(i+1)θ(i+1), i = 1, . . . , n − 1.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Permutations and nested sequences of principal submatrices

Definition We say that an n×n matrix A has a nested sequence of positive prin- cipal minors or simply a nest, if there is a permutation (i1, . . . , in)

• f the set of indices [n] such that

A i1 . . . ij i1 . . . ij

• > 0

j = 1, . . . , n. aθ(1)θ(1),

• aθ(1)θ(1)

aθ(1)θ(2) aθ(2)θ(1) aθ(2)θ(2)

• , . . . ,
• aθ(1)θ(1)

. . . aθ(1)θ(n) . . . . . . . . . aθ(n)θ(1) . . . aθ(n)θ(n)

• Olga Kushel

ON MATRIX D-STABILITY AND RELATED PROPERTIES

Dθ-stability

Definition We call a matrix A D-stable with respect to the direction θ or Dθ- stable if the matrix DA is positive stable for every θ-ordered positive diagonal matrix D.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Dθ-stability

Definition We call a matrix A D-stable with respect to the direction θ or Dθ- stable if the matrix DA is positive stable for every θ-ordered positive diagonal matrix D. Observation (K., 2015) A matrix A is D-stable if it is Dθ-stable for all the possible permu- tations θ of the set [n].

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

P- and Q-matrices: introduction

Definition An n × n matrix A is called a P-matrix if all its principal minors are positive, i.e the inequality A i1 . . . ik i1 . . . ik

• > 0 holds for all

(i1, . . . , ik), 1 ≤ i1 < . . . < ik ≤ n, and all k, 1 ≤ k ≤ n.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

P- and Q-matrices: introduction

Definition An n × n matrix A is called a P-matrix if all its principal minors are positive, i.e the inequality A i1 . . . ik i1 . . . ik

• > 0 holds for all

(i1, . . . , ik), 1 ≤ i1 < . . . < ik ≤ n, and all k, 1 ≤ k ≤ n. Theorem (Fiedler, Pt´ ak, 1962) The following properties of a matrix A are equivalent:

1 All principal minors of A are positive. 2 Every real eigenvalue of A as well as of each principal submatrix

• f A is positive.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

P- and Q-matrices: introduction

Example A =   9 −1 1 0.5 1 1 1 1 3   ; A(2) =         A 1 2 1 2

• A

1 2 1 3

• A

1 2 2 3

• A

1 3 1 2

• A

1 3 1 3

• A

1 3 2 3

• A

2 3 1 2

• A

2 3 1 3

• A

2 3 2 3

• 

       = =   9.5 8.5 −2 10 26 −4 −0.5 0.5 2   ; A(3) = det(A) = 18.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

P- and Q-matrices: introduction

Definition A matrix A is called a Q-matrix if the inequality

• (i1,...,ik)

A i1 . . . ik i1 . . . ik

• > 0

holds for all k, 1 ≤ k ≤ n.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

P- and Q-matrices: introduction

Definition A matrix A is called a Q-matrix if the inequality

• (i1,...,ik)

A i1 . . . ik i1 . . . ik

• > 0

holds for all k, 1 ≤ k ≤ n. Theorem (Hershkowitz, 1983) A set {λ1, . . . , λn}, λi ∈ C, is a spectrum of some P-matrix if and

• nly if it is a spectrum of some Q-matrix.

Corollary Every real eigenvalue of a Q-matrix A is positive.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P-matrices: open problems

Definition A matrix A is called sign-symmetric if the inequality A i1 . . . ik j1 . . . jk

• A

j1 . . . jk i1 . . . ik

• ≥ 0

holds for all sets of indices (i1, . . . , ik), (j1, . . . , jk), where 1 ≤ i1 < . . . < ik ≤ n, 1 ≤ j1 < . . . < jk ≤ n.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P-matrices: open problems

Definition A matrix A is called sign-symmetric if the inequality A i1 . . . ik j1 . . . jk

• A

j1 . . . jk i1 . . . ik

• ≥ 0

holds for all sets of indices (i1, . . . , ik), (j1, . . . , jk), where 1 ≤ i1 < . . . < ik ≤ n, 1 ≤ j1 < . . . < jk ≤ n. Theorem (Carlson, 1973) A sign-symmetric P-matrix is positively stable.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P-matrices: open problems

Definition A matrix A is called sign-symmetric if the inequality A i1 . . . ik j1 . . . jk

• A

j1 . . . jk i1 . . . ik

• ≥ 0

holds for all sets of indices (i1, . . . , ik), (j1, . . . , jk), where 1 ≤ i1 < . . . < ik ≤ n, 1 ≤ j1 < . . . < jk ≤ n. Theorem (Carlson, 1973) A sign-symmetric P-matrix is positively stable. A is a sign-symmetric P-matrix ⇒ A2 is a P-matrix. A is a sign-symmetric P-matrix ⇒ (DA)2 is a P-matrix for every positive diagonal matrix D.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P-matrices: open problems

✻ ✲

λ1(DA) λ2(DA) λ2(A) λ1(A)

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P-matrices: open problems

✻ ✲

λ1(DA) λ2(DA) λ2(A) λ1(A)

■ ✮

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P-matrices: open problems

✻ ✲

λ1(DA) λ2(DA) λ2(A) λ1(A)

■ ✮ ✻

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P-matrices: open problems

✻ ✲

λ1(DA) λ2(DA) λ2(A) λ1(A)

■ ✮ ✻

λ2(A) λ1(A)

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P-matrices: open problems

Definition A matrix A is called a P2-matrix (Q2-matrix) if A and A2 are both P- (respectively, Q-) matrices.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P-matrices: open problems

Definition A matrix A is called a P2-matrix (Q2-matrix) if A and A2 are both P- (respectively, Q-) matrices. Lemma (K., 2013) Let A be a P2-matrix. Then the following matrices are also P2- matrices.

• 1. AT (the transpose of A);
• 2. A−1 (the inverse of A);
• 3. DAD−1, where D is an invertible diagonal matrix;
• 4. PAP−1, where P is a permutation matrix.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P-matrices: open problems

Question (Hershkowitz, Keller, 2003) Are P2-matrices positive stable?

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stabilization by a diagonal matrix

a11, A 1 2 1 2

• , A

1 2 3 1 2 3

• , . . . , A

1 2 . . . n 1 2 . . . n

• .

Definition An n×n positive diagonal matrix D(A) is called a stabilization matrix for an n × n matrix A if all the eigenvalues of D(A)A are positive and simple.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stabilization by a diagonal matrix

a11, A 1 2 1 2

• , A

1 2 3 1 2 3

• , . . . , A

1 2 . . . n 1 2 . . . n

• .

Definition An n×n positive diagonal matrix D(A) is called a stabilization matrix for an n × n matrix A if all the eigenvalues of D(A)A are positive and simple. Definition An n × n real matrix A is called stabilizable if there is at least one stabilization matrix D(A).

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stabilization by a diagonal matrix

Theorem (Fisher, Fuller, 1958) Let A be an n × n real matrix, all of whose leading principal minors are positive. Then there is an n × n positive diagonal matrix D(A), such that all of the eigenvalues of D(A)A are positive and simple.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stabilization by a diagonal matrix

Theorem (Fisher, Fuller, 1958) Let A be an n × n real matrix, all of whose leading principal minors are positive. Then there is an n × n positive diagonal matrix D(A), such that all of the eigenvalues of D(A)A are positive and simple. Definition We say that an n×n matrix A has a nested sequence of positive prin- cipal minors or simply a nest, if there is a permutation (i1, . . . , in)

• f the set of indices [n] such that

A i1 . . . ij i1 . . . ij

• > 0

j = 1, . . . , n. Corollary An n × n real matrix A is stabilizable if it has at least one nested sequence of positive principal minors.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stabilization by a diagonal matrix

Lemma (K., 2014) Let A be an n × n matrix with positive leading principal minors. Then it is stabilizable and the following statements hold:

• 1. We can choose the stabilization matrix D(A) in the following

form D(A) = diag{ǫ1, ǫ2, . . . , ǫn}, where 1 = ǫ1 > ǫ2 > . . . > ǫn > 0.

• 2. There is a stabilization matrix

D0

(A) = diag{ǫ0 1, ǫ0 2, . . . , ǫ0 n},

such that any positive diagonal matrix D(A) = diag{ǫ1, ǫ2, . . . , ǫn} satisfying ǫ1 = ǫ0

1 and ǫi ǫi+1 ≥ ǫ0

i

ǫ0

i+1

(i = 1, . . . , n − 1) is also a stabilization matrix for A.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P2-matrices

Definition An n × n matrix A is called strictly row square diagonally dominant for every order of minors if the following inequalities hold:

• A

α α 2 >

• α,β⊂[n],α=β
• A

α β 2 for any α = (i1, . . . , ik), β = (j1, . . . , jk) and all k = 1, . . . , n. A matrix A is called strictly column square diagonally dominant if AT is strictly row square diagonally dominant.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P2-matrices

Definition An n × n matrix A is called strictly row square diagonally dominant for every order of minors if the following inequalities hold:

• A

α α 2 >

• α,β⊂[n],α=β
• A

α β 2 for any α = (i1, . . . , ik), β = (j1, . . . , jk) and all k = 1, . . . , n. A matrix A is called strictly column square diagonally dominant if AT is strictly row square diagonally dominant. Theorem (Tang et al., 2007) Let A be a P-matrix. If A is strictly row (column) diagonally dom- inant for every order of minors, then A is positive stable.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P2-matrices

Definition An n × n matrix A is called anti-sign symmetric if the following inequalities hold: A α β

• A

β α

• ≤ 0

for any α, β ⊂ {1, . . . , n}, α = β, |α| = |β|.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P2-matrices

Definition An n × n matrix A is called anti-sign symmetric if the following inequalities hold: A α β

• A

β α

• ≤ 0

for any α, β ⊂ {1, . . . , n}, α = β, |α| = |β|. Theorem (Hershkowitz, Keller, 2003) Let A be an anti-sign symmetric P-matrix such that A2 is a P+

0 -

• matrix. Then A is positive stable.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P2-matrices

Theorem (K., 2014) Let an n × n P-matrix A also be a Q2-matrix. Let A have a nested sequence of principal submatrices each of which is also a Q2-matrix. Then A is positive stable.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Stability of P2-matrices

Theorem (K., 2014) Let an n × n P-matrix A also be a Q2-matrix. Let A have a nested sequence of principal submatrices each of which is also a Q2-matrix. Then A is positive stable. Corollaries

• 1. Theorem (Carlson)
• 2. Theorem (Tang et al.)
• 3. Theorem (Hershkowitz, Keller)

Examples

• 1. Hermitian positive definite matrices.
• 2. Strictly totally positive matrices and their generalizations.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

D-stability and Dθ-stability of P2-matrices

Theorem (K., 2015) Let an n × n P-matrix A also be a Q2-matrix. Let A have a nested sequence of principal submatrices each of which is also a Q2-matrix. Then A is Dθ-stable with respect to the permutation θ defined by the given above nested sequence.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

D-stability and Dθ-stability of P2-matrices

Theorem (K., 2015) Let an n × n P-matrix A also be a Q2-matrix. Let A have a nested sequence of principal submatrices each of which is also a Q2-matrix. Then A is Dθ-stable with respect to the permutation θ defined by the given above nested sequence. Theorem (K., 2015) Let an n × n P-matrix A also be a Q2-matrix. Let any principal submatrix of A be a Q2-matrix. Then A is D-stable.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Rank one perturbations of singular M-matrices

Conjecture (Bierkens, Ran, 2014) Suppose H be an n × n symmetric (entry-wise) nonnegative matrix with geometrically simple eigenvalue ρ(H). Let u, v ∈ Rn be (entry- wise) positive. Then A := ρ(H)I − H + u ⊗ vT is D-stable.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Rank one perturbations of singular M-matrices

Conjecture (Bierkens, Ran, 2014) Suppose H be an n × n symmetric (entry-wise) nonnegative matrix with geometrically simple eigenvalue ρ(H). Let u, v ∈ Rn be (entry- wise) positive. Then A := ρ(H)I − H + u ⊗ vT is D-stable. Theorem (K., 2015) Suppose H be an n × n sign-symmetric irreducible nonnegative matrix. Let u, v ∈ Rn be (entry-wise) positive. Then A := ρ(H)I − H + u ⊗ vT is D-stable.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

Rank one perturbations of singular M-matrices

Lemma (K., 2015) Suppose H be an n × n sign-symmetric irreducible nonnegative matrix. Let u, v ∈ Rn be (entry-wise) positive. Then A := ρ(H)I−H+u⊗vT is a Q2-matrix as well as any principal submatrix

• f A.

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES

THANK YOU!

Olga Kushel ON MATRIX D-STABILITY AND RELATED PROPERTIES