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

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

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

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

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

Outline 1 D -stable matrices 2 Permutations and nested sequences of principal submatrices 3 D θ -stability 4 P - and Q -matrices: introduction 5 Stability of P -matrices: open problems 6 Stabilization by a diagonal matrix 7 Stability of P 2 -matrices 8 D -stability and D θ -stability of P 2 -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 { d 11 , . . . , d nn } 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 d θ ( i ) θ ( i ) ≥ d θ ( i +1) θ ( i +1) , i = 1 , . . . , n − 1 . We call the matrix D strictly θ -ordered if d θ ( i ) θ ( i ) > d θ ( 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 ( i 1 , . . . , i n ) of the set of indices [ n ] such that � i 1 � . . . i j > 0 j = 1 , . . . , n . A i 1 . . . i j � � a θ (1) θ (1) . . . a θ (1) θ ( n ) � � � � a θ (1) θ (1) a θ (1) θ (2) � . . . � � � . . . a θ (1) θ (1) , � , . . . , � � . . . � � a θ (2) θ (1) a θ (2) θ (2) � � � � � a θ ( n ) θ (1) a θ ( n ) θ ( n ) . . . � � Olga Kushel ON MATRIX D -STABILITY AND RELATED PROPERTIES

D θ -stability Definition We call a matrix A D -stable with respect to the direction θ or D θ - 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

D θ -stability Definition We call a matrix A D -stable with respect to the direction θ or D θ - 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 D θ -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 � i 1 � . . . i k are positive, i.e the inequality A > 0 holds for all i 1 . . . i k ( i 1 , . . . , i k ) , 1 ≤ i 1 < . . . < i k ≤ 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 � i 1 � . . . i k are positive, i.e the inequality A > 0 holds for all i 1 . . . i k ( i 1 , . . . , i k ) , 1 ≤ i 1 < . . . < i k ≤ 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 of A is positive. Olga Kushel ON MATRIX D -STABILITY AND RELATED PROPERTIES

P - and Q -matrices: introduction Example   9 − 1 1  ; A = 0 . 5 1 1  1 1 3 � 1 � � 1 � � 1 �  2 2 2  A A A 1 2 1 3 2 3    � 1 � � 1 � � 1 �  3 3 3 A (2) =   A A A =   1 2 1 3 2 3     � 2 � � 2 � � 2 � 3 3 3   A A A 1 2 1 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 � i 1 � . . . i k � > 0 A i 1 . . . i k ( i 1 ,..., i k ) 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 � i 1 � . . . i k � > 0 A i 1 . . . i k ( i 1 ,..., i k ) holds for all k , 1 ≤ k ≤ n . Theorem (Hershkowitz, 1983) A set { λ 1 , . . . , λ n } , λ i ∈ C , is a spectrum of some P -matrix if and only 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 � i 1 � � j 1 � . . . i k . . . j k ≥ 0 A A j 1 . . . j k i 1 . . . i k holds for all sets of indices ( i 1 , . . . , i k ) , ( j 1 , . . . , j k ), where 1 ≤ i 1 < . . . < i k ≤ n , 1 ≤ j 1 < . . . < j k ≤ 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 � i 1 � � j 1 � . . . i k . . . j k ≥ 0 A A j 1 . . . j k i 1 . . . i k holds for all sets of indices ( i 1 , . . . , i k ) , ( j 1 , . . . , j k ), where 1 ≤ i 1 < . . . < i k ≤ n , 1 ≤ j 1 < . . . < j k ≤ 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 � i 1 � � j 1 � . . . i k . . . j k ≥ 0 A A j 1 . . . j k i 1 . . . i k holds for all sets of indices ( i 1 , . . . , i k ) , ( j 1 , . . . , j k ), where 1 ≤ i 1 < . . . < i k ≤ n , 1 ≤ j 1 < . . . < j k ≤ n . Theorem (Carlson, 1973) A sign-symmetric P -matrix is positively stable. A is a sign-symmetric P -matrix ⇒ A 2 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 ✻ λ 1 ( A ) ✲ λ 1 ( DA ) λ 2 ( DA ) λ 2 ( A ) Olga Kushel ON MATRIX D -STABILITY AND RELATED PROPERTIES

Stability of P -matrices: open problems ✻ λ 1 ( A ) ■ ✲ λ 1 ( DA ) λ 2 ( DA ) ✮ λ 2 ( A ) Olga Kushel ON MATRIX D -STABILITY AND RELATED PROPERTIES

Stability of P -matrices: open problems ✻ ✻ λ 1 ( A ) ■ ✲ λ 1 ( DA ) λ 2 ( DA ) ✮ λ 2 ( A ) Olga Kushel ON MATRIX D -STABILITY AND RELATED PROPERTIES

Stability of P -matrices: open problems ✻ ✻ λ 1 ( A ) λ 1 ( A ) ■ ✲ λ 1 ( DA ) λ 2 ( DA ) ✮ λ 2 ( A ) λ 2 ( A ) Olga Kushel ON MATRIX D -STABILITY AND RELATED PROPERTIES

Stability of P -matrices: open problems Definition A matrix A is called a P 2 -matrix ( Q 2 -matrix) if A and A 2 are both P - (respectively, Q -) matrices. Olga Kushel ON MATRIX D -STABILITY AND RELATED PROPERTIES