From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Powers of a real matrix Date: 11 Jan 2001 12:36:10 -0500 Newsgroups: sci.math Summary: Spectral radius as a matrix norm? In article <3A5DCEB8.CE8A8232@penmaen.demon.co.uk>, Dr. Uwe Prells wrote: :Hello Ben, : :Let's take the spectral norm for ||A||, that is max |lambda(A)|. (!!!!!!!) Correction: The name for max |lambda(A)| is "spectral radius" and it is not a norm at all. To see this, consider the non-zero matrix [ 0 1 ] [ 0 0 ] which has spectral radius 0. Also, triangle inequality can be easily violated by counterexamples. (Try it; hint: the sum of two nilpotent matrices need not be nilpotent.) In some textbooks, the expression "spectral norm" is used to denote the maximal singular value of A. This is indeed a norm, and I don't like the use of "spectral" in this context. "Euclidean operator norm" is a clumsy, but more descriptive name. :Since :your bounds c1,c2 are independent of :the power m the condition is that the absolute values of :all eigenvalues of A must be lower or equal to one. (!!!) With some substantial amendments, see below. : :Regards Uwe : :ben_geffen@my-deja.com wrote: : :> Suppose A is an n by n real matrix and the powers :A,A^2,A^3,... are :> bounded above and below i.e there are constants c1 > 0 c2 > 0 :> such that c1 <= ||A^m|| <= c2 for every m>=1 when || || is a :> matrix norm. :> :> Is it possible to characterize such matrices ? :> There has been an unnecessarily passionate flame war over a similar situation last year. Why "unnecessarily"? Because it's a matter mostly of public record, or there is an easy proof based on accessible standard results. (There are many textbooks discussing "Jordan Normal Form" and "Functions of Matrices".) So, let me discuss the two conditions separately: (a) Suppose there exists a constant c such that for every integer m>=1, ||A^m|| <= c. Then (1) every eigenvalue of A has absolute value less than or equal to 1, (2) the eigenvalues with absolute value 1 are simple poles of the resolvent (lambda*I - A)^(-1) , that is, the corresponding (possibly repeated) Jordan cells are one-dimensional. The conjunction of (1) and (2) is both necessary and sufficient for the positive integer powers of A being bounded from above. Proof can be conducted using Jordan canonical form. (b) When are the norms of powers bounded from below by a positive constant (independently of situation (a))? Necessarily at least one of the eigenvalues must have absolute value greater than or equal to 1. (Proof by contradiction). And this condition is also sufficient, in a stronger form: if satisfied, then one of the lower bounds is 1. Then you put (a) and (b) together. Best wishes, ZVK(Slavek). ============================================================================== From: "Dr. Uwe Prells" Subject: Re: Powers of a real matrix Date: Thu, 11 Jan 2001 23:44:30 +0000 Newsgroups: sci.math Hello Joe, Sorry, you are right. To be precise the spectral norm of a matrix A is the positive square root of the the maximum eigenvalue of A'*A where the prime indicates conjugation and transposition. Joe Taylor wrote: > Dr. Uwe Prells wrote: > > the condition is that the absolute values of all eigenvalues > > of A must be lower or equal to one. > > Not quite. Consider the 2x2 triangular matrix with (repeated) > eigenvalue 1 > > A = 1 1 > 0 1 > ||A|| = sqrt((3+sqrt(5) )/2) >1 hence A^m unbound with increasing m.. Regards Uwe > > It easy to see (by induction) that > > A^m = 1 m > 0 1 > > so ||A^m|| grows without bound as m increases. > > > Let's take the spectral norm for ||A||, that is max | lambda(A)|. > > The spectral *radius* [max | lambda(A)| ] is not a matrix norm > > > > ben_geffen@my-deja.com wrote: > > > > Suppose A is an n by n real matrix and the powers A,A^2,A^3,... are > > > bounded above and below i.e there are constants c1 > 0 c2 > 0 such > >> that > > > c1 <= ||A^m|| <= c2 for every m>=1 when || || is a matrix norm. > > > > > > Is it possible to characterize such matrices ? > > -- > > mailto:joe_taylor@mistral.co.uk ============================================================================== From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Powers of a real matrix Date: 11 Jan 2001 21:13:54 -0500 Newsgroups: sci.math In article <93lko3$6o5$1@nnrp1.deja.com>, wrote: :In article <93k6qg$r15$1@nnrp1.deja.com>, : ben_geffen@my-deja.com wrote: :> Suppose A is an n by n real matrix and the powers :> A,A^2,A^3,... are bounded above and below i.e there are :> constants c1 > 0 c2 > 0 such that :> c1 <= ||A^m|| <= c2 for every m>=1 when || || is a matrix :> norm. :> :> Is it possible to characterize such matrices ? :> : :If your matrix has these properties, then so does its Jordan :canonical form. : :This immediately implies: :The *complex* Jordan canonical form is strictly diagonal. ... too strict. Only eigenvalues on the boundary of the unit disk have to generate a diagonal block. The Jordan structure corresponding to the interior eigenvalues is not restricted. Example: [ 0 1 0 ] [ 0 0 0 ] [ 0 0 -1 ] :The maximum modulus of the eigenvalues must be 1. That's correct. Cheers, ZVK(Slavek).