Perturbation Bounds for Determinants and Characteristic Polynomials

Transcription

1 Perturbation Bounds for Determinants and Characteristic Polynomials Ilse Ipsen North Carolina State University, Raleigh, USA Joint work with: Rizwana Rehman

2 Characteristic Polynomials n n complex matrix A det(λi A) = λ n + c 1 λ n c n 1 λ + c n c 1 = trace(a) c n = ( 1) n det(a) Application (Dean Lee, NCSU) Thermodynamic properties of systems of fermions Z k ( 1) k c k is partition function

3 Perturbation Bounds n n complex matrices A and A + E det(λi A) = λ n + c 1 λ n c n 1 λ + c n det(λi (A + E)) = λ n + c 1 λ n c n 1 λ + c n Trace: c 1 c 1 = trace(a+e) trace(a) = trace(e) n E 2 Determinant: c n c n = det(a + E) det(a)? Other coefficients: Bound c k c k by means of results for determinants

4 Existing Bounds for Determinants Friedland 1982, Bhatia 1987 det(a+e) det(a) n max{ A p, A + E p } n 1 E p Proof: Fréchet derivatives of wedge products Godunov, Antonov, Kiriljuk & Kostin 1988 If n A 1 2 E 2 < 1 then det(a + E) det(a) det(a) n A n A 1 2 E 2 E 2 Our idea: Use determinant expansions of diagonal matrices

5 Determinant Expansion of Diagonal Matrices 1 A = σ σ 2 σ 3 e 11 e 12 e 13 E = e 21 e 22 e 23 e 31 e 32 e 33 Expansion: det(a + E) = det(a) + S 1 + S 2 + det(e) where ( ) e22 e S 1 = σ 1 det 23 +σ e 32 e 2 det 33 ( e11 e 13 e 31 e 33 ) +σ 3 det ( ) e11 e 12 e 21 e 22 S 2 = σ 1 σ 2 e 33 + σ 1 σ 3 e 22 + σ 2 σ 3 e 11

6 Determinant Expansion of Diagonal Matrices A = σ 1... Expansion: σ n det(a + E) = det(a) + S S n 1 + det(e) where S k = σ i1 σ ik det(e i1...i k ) 1 i 1 < <i k n σ i1 σ ik det(e i1...i k ) product of k diagonal elements of A principal minor of order n k of E

7 Determinant Bounds for Diagonal Matrices A = diag ( σ 1... σ n ) σ 1... σ n 0 det(a + E) det(a) S S n 1 + det(e) where S k σ i1 σ ik det(e i1...i k ) 1 i 1 < <i k n Hadamard s inequality: det(e i1...i k ) E i1...i k n k E n k (2 norm)

8 Elementary Symmetric Functions S k σ i1 σ ik E n k 1 i 1 < <i k n kth elementary symmetric function of σ 1,..., σ n s k σ i1 σ ik 1 i 1 < <i k n If n = 3 then s 1 = σ 1 + σ 2 + σ 3 s 2 = σ 1 σ 2 + σ 1 σ 3 + σ 2 σ 1 + σ 2 σ 3 + σ 3 σ 1 + σ 3 σ 2 s 3 = σ 1 σ 2 σ 3 S k s k E n k

9 Determinant Bound for Diagonal Matrices Non negative diagonal matrix A = diag ( σ 1... σ n ) σ 1... σ n 0 Elementary symmetric functions of σ 1,..., σ n s k σ i1 σ ik 1 i 1 < <i k n Determinant perturbation bound det(a) det(a + E) s n 1 E + + s 1 E n 1 + E n First order bound det(a) det(a + E) s n 1 E + O ( E 2) where s n 1 n σ 1... σ n 1

10 Determinant Bound for General Matrices Complex n n matrix A with singular values σ 1... σ n 0 Elementary symmetric functions of singular values s k σ i1 σ ik 1 i 1 < <i k n Determinant perturbation bound det(a) det(a + E) s n 1 E + + s 1 E n 1 + E n First order bound det(a) det(a + E) s n 1 E + O ( E 2) where s n 1 n σ 1... σ n 1

11 Derivative of Determinant If x is a real scalar then d dx det(a + xe) x=0 s n 1 E where s n 1 n σ 1... σ n 1 Local (absolute) condition number for determinant: (n 1)st elementary symmetric function of singular values

12 Relative Bound If A is nonsingular then 1/det(A) = det(a 1 ) Determinant expansion of I + A 1 E det(a + E) det(a) det(a) = S S n 1 + det(a 1 E) Minors S k det((a 1 E) i1...i k ) 1 i 1 < <i k n Relative perturbation bound det(a + E) det(a) det(a) ( A 1 E + 1) n 1

13 Coefficients of Characteristic Polynomial n n complex matrices A and A + E det(λi A) = λ n + c 1 λ n c n 1 λ + c n det(λi (A + E)) = λ n + c 1 λ n c n 1 λ + c n Friedland 1982, Bhatia 1987 ( ) n c k c k k max{ A p, A + E p } k 1 E p k Proof: Fréchet derivatives of wedge products

14 A Perturbed Coefficient c k = ( 1) k 1 i 1 <...<i n k n det(a i1...i n k + E i1...i n k ) Sum of ( n k) minors Each minor is determinant of matrix of order k Apply determinant bound to each term Bounds contain k largest singular values σ 1,..., σ k

15 Bound for Perturbed Coefficient Elementary symmetric function of k largest singular values σ 1,..., σ k s (k) j = σ i1... σ ij 1 i 1 <...<i j k Perturbation bound ( ) n ( c k c k s (k) k 1 E + + s(k) 1 k E k 1 + E k) First order bound ( ) n c k c k s (k) k 1 k E + O( E 2 ) where s (k) k 1 k σ 1... σ k 1

16 Normal (or Hermitian) Matrices Perturbation bound c k c k (n k + 1)s k 1 E + ( ) ( ) n 1 n + s 1 E k 1 + E k k 1 k Elementary functions in terms of all singular values (not just the largest ones) Binomial factors smaller than in general bound First order bound c k c k (n k + 1)s k 1 E + O ( E 2) Normal (or Hermitian) matrices may have better conditioned coefficients

17 Summary Characteristic polynomial of n n complex matrix A det(λi A) = λ n + c 1 λ n c n 1 λ + c n c k elementary symmetric functions of eigenvalues Perturbation bounds in terms of elementary symmetric functions of singular values Absolute local (first order) condition numbers Normal (or Hermitian) matrices may have better conditioned coefficients