# The ε-pseudospectrum of a Matrix

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1 The ε-pseudospectrum of a Matrix Feb 16, 2015 () The ε-pseudospectrum of a Matrix Feb 16, / 18

2 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of Problems () The ε-pseudospectrum of a Matrix Feb 16, / 18

3 Let A M n (C) matrix. () The ε-pseudospectrum of a Matrix Feb 16, / 18

4 Let A M n (C) matrix. Spectrum of A σ(a) = {λ C det(λi A) = 0} () The ε-pseudospectrum of a Matrix Feb 16, / 18

5 Let A M n (C) matrix. Spectrum of A σ(a) = {λ C det(λi A) = 0} Positive Semi-definite matrix M = M and σ(m) R + () The ε-pseudospectrum of a Matrix Feb 16, / 18

6 Let A M n (C) matrix. Spectrum of A σ(a) = {λ C det(λi A) = 0} Positive Semi-definite matrix M = M and σ(m) R + M = AA () The ε-pseudospectrum of a Matrix Feb 16, / 18

7 Let A M n (C) matrix. Spectrum of A σ(a) = {λ C det(λi A) = 0} Positive Semi-definite matrix M = M and σ(m) R + M = AA Singular values of A { s s σ(aa )} () The ε-pseudospectrum of a Matrix Feb 16, / 18

8 Let A M n (C) matrix. Spectrum of A σ(a) = {λ C det(λi A) = 0} Positive Semi-definite matrix M = M and σ(m) R + M = AA Singular values of A { s s σ(aa )} s i (A) i th largest singular value of A () The ε-pseudospectrum of a Matrix Feb 16, / 18

9 Let A M n (C) matrix. Spectrum of A σ(a) = {λ C det(λi A) = 0} Positive Semi-definite matrix M = M and σ(m) R + M = AA Singular values of A { s s σ(aa )} s i (A) i th largest singular value of A Spectral Norm of A A = max x =1 Ax 2 = s 1 (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

10 Let A M n (C) matrix. Spectrum of A σ(a) = {λ C det(λi A) = 0} Positive Semi-definite matrix M = M and σ(m) R + M = AA Singular values of A { s s σ(aa )} s i (A) i th largest singular value of A Spectral Norm of A A = max x =1 Ax 2 = s 1 (A) Note: The spectrum of A is an unstable. Invertibility is affected quickly even by small perturbations. Some generalizations of the spectrum are 1 numerical range 2 polynomial numerical hull 3 pseudospectrum () The ε-pseudospectrum of a Matrix Feb 16, / 18

11 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of Problems () The ε-pseudospectrum of a Matrix Feb 16, / 18

12 The ε Pseudospectrum Let A M n (C), a norm on M n (C), and ε > 0 () The ε-pseudospectrum of a Matrix Feb 16, / 18

13 The ε Pseudospectrum Let A M n (C), a norm on M n (C), and ε > 0 σ ε (A) = {z C z σ(a + E) for some E with E < ε} σ ε (A) = {z C (z A)v < ε for some v with v = 1} σ ε (A) = {z C (zi A) 1 1 ε } () The ε-pseudospectrum of a Matrix Feb 16, / 18

14 The ε Pseudospectrum Let A M n (C), a norm on M n (C), and ε > 0 σ ε (A) = {z C z σ(a + E) for some E with E < ε} σ ε (A) = {z C (z A)v < ε for some v with v = 1} σ ε (A) = {z C (zi A) 1 1 ε } If is the spectral norm, then σ ε (A) = {z C s n (zi A) ε} () The ε-pseudospectrum of a Matrix Feb 16, / 18

15 Pseudospectral Radius and Abscissa Pseudospectral Radius ρ ε (A) = max z σ ε z The radius of the smallest circle centered at the origin containing σ ε (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

16 Pseudospectral Radius and Abscissa Pseudospectral Radius ρ ε (A) = max z σ ε z The radius of the smallest circle centered at the origin containing σ ε (A) Pseudospectral Abscissa ρ ε (A) = max z σ ε Re(z) The rightmost vertical support line of σ ε (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

17 Goal To completely describe the geometry of the pseudospectrum of a matrix. () The ε-pseudospectrum of a Matrix Feb 16, / 18

18 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of Problems () The ε-pseudospectrum of a Matrix Feb 16, / 18

19 Reducing Properties () The ε-pseudospectrum of a Matrix Feb 16, / 18

20 Reducing Properties ) 1 σ ε (diag(d 1,, d n ) () The ε-pseudospectrum of a Matrix Feb 16, / 18

21 Reducing Properties ) 1 σ ε (diag(d 1,, d n ) = n D(d i, ε) i=1 () The ε-pseudospectrum of a Matrix Feb 16, / 18

22 Reducing Properties ) 1 σ ε (diag(d 1,, d n ) = n D(d i, ε) i=1 2 σ ε (UAU ) = σ ε (A) for any unitary U () The ε-pseudospectrum of a Matrix Feb 16, / 18

23 Reducing Properties ) 1 σ ε (diag(d 1,, d n ) = n D(d i, ε) i=1 2 σ ε (UAU ) = σ ε (A) for any unitary U 3 If A is normal, i.e. AA = A A, then σ ε (A) = z σ(a) D(z, ε) () The ε-pseudospectrum of a Matrix Feb 16, / 18

24 Reducing Properties ) 1 σ ε (diag(d 1,, d n ) = n D(d i, ε) i=1 2 σ ε (UAU ) = σ ε (A) for any unitary U 3 If A is normal, i.e. AA = A A, then σ ε (A) = 4 σ ε (A 1 A 2 ) = σ ε (A 1 ) σ ε (A 2 ) z σ(a) D(z, ε) () The ε-pseudospectrum of a Matrix Feb 16, / 18

25 Reducing Properties ) 1 σ ε (diag(d 1,, d n ) = n D(d i, ε) i=1 2 σ ε (UAU ) = σ ε (A) for any unitary U 3 If A is normal, i.e. AA = A A, then σ ε (A) = 4 σ ε (A 1 A 2 ) = σ ε (A 1 ) σ ε (A 2 ) z σ(a) D(z, ε) 5 σ ε (αi + βa) = α + βσ ε β (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

26 Reducing Properties ) 1 σ ε (diag(d 1,, d n ) = n D(d i, ε) i=1 2 σ ε (UAU ) = σ ε (A) for any unitary U 3 If A is normal, i.e. AA = A A, then σ ε (A) = 4 σ ε (A 1 A 2 ) = σ ε (A 1 ) σ ε (A 2 ) z σ(a) D(z, ε) 5 σ ε (αi + βa) = α + βσ ε β (A) 6 σ ε (A ) = σ ε (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

27 Containment Properties 1 σ(a) σ ε (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

28 Containment Properties 1 σ(a) σ ε (A) 2 If ε 1 ε 2, then σ ε1 (A) σ ε2 (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

29 Containment Properties 1 σ(a) σ ε (A) 2 If ε 1 ε 2, then σ ε1 (A) σ ε2 (A) 3 σ ε E (A) σ ε (A + E) σ ε+ E (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

30 Containment Properties 1 σ(a) σ ε (A) 2 If ε 1 ε 2, then σ ε1 (A) σ ε2 (A) 3 σ ε E (A) σ ε (A + E) σ ε+ E (A) 4 If κ(x ) = X X 1 = s1(x ) s n(x ), then σ ε κ(s) (SAS 1 ) σ ε (A) σ εκ(s) (SAS 1 ) () The ε-pseudospectrum of a Matrix Feb 16, / 18

31 Containment Properties 1 σ(a) σ ε (A) 2 If ε 1 ε 2, then σ ε1 (A) σ ε2 (A) 3 σ ε E (A) σ ε (A + E) σ ε+ E (A) 4 If κ(x ) = X X 1 = s1(x ) s n(x ), then σ ε κ(s) (SAS 1 ) σ ε (A) σ εκ(s) (SAS 1 ) 5 If u, v are unit vectors such that X v = λv and u X = λu, then κ(λ) = 1/ u v. If A has n distinct eigenvalues, then σ ε (A) D(λ, εκ(λ)) λ σ(a) () The ε-pseudospectrum of a Matrix Feb 16, / 18

32 Containment Properties 1 σ ε (A) λ W (A) D(λ, ε) () The ε-pseudospectrum of a Matrix Feb 16, / 18

33 Containment Properties 1 σ ε (A) D(λ, ε) λ W (A) 2 σ ε (A) D(λ, ε + dep(a)) λ σ(a) () The ε-pseudospectrum of a Matrix Feb 16, / 18

34 Containment Properties 1 σ ε (A) D(λ, ε) λ W (A) 2 σ ε (A) D(λ, ε + dep(a)) λ σ(a) 3 If A = [a ij ] and r j = n a jk, then k=1 j n σ ε (A) D(a jj, r j + ε n) j=1 () The ε-pseudospectrum of a Matrix Feb 16, / 18

35 Geometric Properties 1 σ ε (A) has at most n connected components () The ε-pseudospectrum of a Matrix Feb 16, / 18

36 Geometric Properties 1 σ ε (A) has at most n connected components 2 Each connected component contains at least one eigenvalue of A () The ε-pseudospectrum of a Matrix Feb 16, / 18

37 Geometric Properties 1 σ ε (A) has at most n connected components 2 Each connected component contains at least one eigenvalue of A [ ] xi A ɛi 3 z = x + iy σ ε (A) if and only if iy is an eigenvalue of ɛi A xi () The ε-pseudospectrum of a Matrix Feb 16, / 18

38 Geometric Properties 1 σ ε (A) has at most n connected components 2 Each connected component contains at least one eigenvalue of A [ ] xi A ɛi 3 z = x + iy σ ε (A) if and only if iy is an eigenvalue of ɛi A xi 4 σ ε (A) has no flat portions () The ε-pseudospectrum of a Matrix Feb 16, / 18

39 Geometric Properties 1 σ ε (A) has at most n connected components 2 Each connected component contains at least one eigenvalue of A [ ] xi A ɛi 3 z = x + iy σ ε (A) if and only if iy is an eigenvalue of ɛi A xi 4 σ ε (A) has no flat portions 5 σ ε (A) generally not convex, nor simply-connected () The ε-pseudospectrum of a Matrix Feb 16, / 18

40 Geometric Properties 1 σ ε (A) has at most n connected components 2 Each connected component contains at least one eigenvalue of A [ ] xi A ɛi 3 z = x + iy σ ε (A) if and only if iy is an eigenvalue of ɛi A xi 4 σ ε (A) has no flat portions 5 σ ε (A) generally not convex, nor simply-connected 6 If σ ε (A) = n D(z i, ε) such that D(z j, ε) D(z i, ε) for all j. Then A is normal. i=1 i=1 j () The ε-pseudospectrum of a Matrix Feb 16, / 18

41 Geometric Properties 1 σ ε (A) has at most n connected components 2 Each connected component contains at least one eigenvalue of A [ ] xi A ɛi 3 z = x + iy σ ε (A) if and only if iy is an eigenvalue of ɛi A xi 4 σ ε (A) has no flat portions 5 σ ε (A) generally not convex, nor simply-connected 6 If σ ε (A) = n D(z i, ε) such that D(z j, ε) D(z i, ε) for all j. Then A is normal. i=1 i=1 j 7 If σ ε (A) = σ ε (B) for any ε, then σ ε (A) and σ ε (B) have the same minimal polynomial. () The ε-pseudospectrum of a Matrix Feb 16, / 18

42 Geometric Properties 1 σ ε (A) has at most n connected components 2 Each connected component contains at least one eigenvalue of A [ ] xi A ɛi 3 z = x + iy σ ε (A) if and only if iy is an eigenvalue of ɛi A xi 4 σ ε (A) has no flat portions 5 σ ε (A) generally not convex, nor simply-connected 6 If σ ε (A) = n D(z i, ε) such that D(z j, ε) D(z i, ε) for all j. Then A is normal. i=1 i=1 j 7 If σ ε (A) = σ ε (B) for any ε, then σ ε (A) and σ ε (B) have the same minimal polynomial. Converse is not true. () The ε-pseudospectrum of a Matrix Feb 16, / 18

43 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of Problems () The ε-pseudospectrum of a Matrix Feb 16, / 18

44 2 2 Matrix Reduction () The ε-pseudospectrum of a Matrix Feb 16, / 18

45 2 2 Matrix Reduction [ ] a b Let A = with σ(a) = {λ 1, λ 2 } c d () The ε-pseudospectrum of a Matrix Feb 16, / 18

46 2 2 Matrix Reduction [ ] a b Let A = with σ(a) = {λ 1, λ 2 } c d = σ ε (A) = (a+d) 2 + σ ε ([ a d 2 b c d a 2 ]) () The ε-pseudospectrum of a Matrix Feb 16, / 18

47 2 2 Matrix Reduction [ ] a b Let A = with σ(a) = {λ 1, λ 2 } c d = σ ε (A) = (a+d) 2 + σ ε ([ a d 2 b c d a 2 ]) ([ã ]) b = α + e iβ σ ε c ã () The ε-pseudospectrum of a Matrix Feb 16, / 18

48 2 2 Matrix Reduction [ ] a b Let A = with σ(a) = {λ 1, λ 2 } c d = σ ε (A) = (a+d) 2 + σ ε ([ a d 2 b c d a 2 ]) = σ ε (A) = σ ε (UAU ) = α + e iβ σ ε ([ r s ([ã ]) b = α + e iβ σ ε c ã 0 r ]), where () The ε-pseudospectrum of a Matrix Feb 16, / 18

49 2 2 Matrix Reduction [ ] a b Let A = with σ(a) = {λ 1, λ 2 } c d = σ ε (A) = (a+d) 2 + σ ε ([ a d 2 b c d a 2 ]) = σ ε (A) = σ ε (UAU ) = α + e iβ σ ε ([ r s ([ã ]) b = α + e iβ σ ε c ã 0 r ]), where α = λ1+λ2 2 and re 2iβ = (λ 1 λ 2 ) 2 and s = tr((a αi )(A αi ) ) 2r 2 () The ε-pseudospectrum of a Matrix Feb 16, / 18

50 2 2 Matrices [ ] r s Let A = 0 r () The ε-pseudospectrum of a Matrix Feb 16, / 18

51 2 2 Matrices [ ] r s Let A = 0 r If r = 0, then () The ε-pseudospectrum of a Matrix Feb 16, / 18

52 2 2 Matrices [ ] r s Let A = 0 r If r = 0, then σ ε (A) = D(0, ε 2 + εs) () The ε-pseudospectrum of a Matrix Feb 16, / 18

53 2 2 Matrices [ ] r s Let A = 0 r If r = 0, then σ ε (A) = D(0, ε 2 + εs) If r 0, then any z = x + iy σ ε (A) satisfies () The ε-pseudospectrum of a Matrix Feb 16, / 18

54 2 2 Matrices [ ] r s Let A = 0 r If r = 0, then σ ε (A) = D(0, ε 2 + εs) If r 0, then any z = x + iy σ ε (A) satisfies y 2 + x 2 + ε2 s 2 4r 2 2 ( ε 2 + r 1 + s2 4r 2 ) y 2 + x 2 + ε2 s 2 4r 2 r 2 ( ε s2 4r 2 ) 0 () The ε-pseudospectrum of a Matrix Feb 16, / 18

55 2 2 Matrices 1 ρ ε (A) = e iθ ρ ε (A 1 ) () The ε-pseudospectrum of a Matrix Feb 16, / 18

56 2 2 Matrices 1 ρ ε (A) = e iθ ρ ε (A 1 ) 2 α ε (A) = tr(a) 2 + α ε (A 0 ) () The ε-pseudospectrum of a Matrix Feb 16, / 18

57 2 2 Matrices 1 ρ ε (A) = e iθ ρ ε (A 1 ) 2 α ε (A) = tr(a) 2 + α ε (A 0 ) Method of Lagrange Multipliers maximize f (x, y) subject to g(x, y) = c Λ(x, y, λ) = f (x, y) + λ(g(x, y) c) = Λ(x, y, λ) = 0 () The ε-pseudospectrum of a Matrix Feb 16, / 18

58 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of Problems () The ε-pseudospectrum of a Matrix Feb 16, / 18

59 Problems 1 A generalized quadratic matrix is of the form [ ] ai bt A = ct di What can we say about the pseudospectral radius/abscissa of A? 2 Consider real structured ε pseudospectrum of A M n (R) σ R ε (A) = {z σ(a + E) E M n (R), E < ε} 3 Consider Special types of nonnormal Matrices. () The ε-pseudospectrum of a Matrix Feb 16, / 18

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