From: mittelmann@asu.edu Subject: Re: Matrix equation Date: Thu, 06 May 1999 02:56:16 GMT Newsgroups: sci.math,sci.math.num-analysis Keywords: Lyapunov matrix equation, Sylvester matrix equation In article <37307EB3.2A91@ThPhys.Uni-Heidelberg.DE>, Herbert Nachbagauer wrote: > Hello world, > I came across a nasty matrix equation > > A = R X + X R^T > > where A and R are given n x n matrices, and I > numerically have to solve the equation for the matrix X. > I found only a pedestrian way to solve this. First, > I arranged the elements of A in vector form > (A_11,A_12, ... A_1n,A_21,A_22, ... ... ... A_nn) > of dimension n^2, and the same for the unknown X. > Then the equation turns into > > A = Q X > > with Q being the n x n x n x n matrix > of block form > > R + 1 R_11 1 R_12 1 R_13 ... > Q = 1 R_21 R + 1 R_22 1 R_23 ... > . . . > . . . > . . R + 1 R_nn > > ( 1 is the n x n unity matrix). > So one 'only' has to invert Q. However, the > dimension is n^4, so even for modest n, > n^4 is quite a big number. Is there any > smart (numeric) algorithm or method that does better > than this naive solution ? > Hi, your problem appears to be the Lyapunov matrix equation, a special case of the Sylvester matrix equation. You find theory in good numerical linear algebra books such as Golub&VanLoan and algorithms, for example, at netlib. Go to http://www.netlib.org and search for Sylvester equation. toms705 is a good starting point. Hans Mittelmann -----------== Posted via Deja News, The Discussion Network ==---------- http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own ============================================================================== From: Didier Henrion Subject: Re: Help with MATRIX EQUATION Date: Fri, 16 Jul 1999 08:30:08 +0200 Newsgroups: sci.math To: "Ashok. R" Ashok. R wrote: > "To derive a stable adaptation law, we first observe that A is > asymptotically stable and there exists a matrix P such that > > A(T)P + PA = -I > > Where A(T) denotes the transpose of A, I is the identity matrix of same > order as A". > > Now I do know what 'asymptotically stable' means - all eigen values have > strictly negative real parts. However, I do not understand the role of > the mysterious matrix, P. If P exists, does it mean anything in terms of > linear algebra concepts ? > > Is P unique? Dear Ashok, your equation is referred to as a Lyapunov equation. Lyapunov's theorem states that matrix A has all its eigenvalues in the left half-plane if and only if there exists a unique symmetric positive definite matrix P (i.e. with positive real eigenvalues) solution to the Lyapunov equation. You will learn more about this in any standard book on linear systems control theory. The best I know is T. Kailath "Linear Systems" Prentice Hall, NY, 1980 Hope this helps, ------------------------------------------------------------------ Didier Henrion Phone: +33 5 61 33 69 49 LAAS-CNRS, Bureau E50 Fax: +33 5 61 33 69 69 7, Avenue du Colonel Roche mailto:henrion@laas.fr 31077 Toulouse, Cedex 4, France http://www.laas.fr/~henrion ------------------------------------------------------------------ ============================================================================== From: Lars Imsland Subject: Re: Geometric sum of matrixes Date: Fri, 08 Oct 1999 08:54:28 +0200 Newsgroups: sci.math "G. A. Edgar" wrote: > > In article , Fredrik Glöckner > wrote: > > > If a is a scalar and |a|<1, we know that > > > > oo 2i 1 > > sum a = ----- > > i=0 1-a^2 > > > > > > Suppose that A is a matrix, and that all eigenvalues of A are less than > > 1 in absolute value. Does there exist a similar formula for the sum: > > > > I + AA' + AAA'A' + AAAA'A'A' + ... > > > > Thanks for any help, > > Formally, it is the solution T of I+ATA'=T. If the series converges, > then it will converge to the (a?) solution of this. If A and A' > commute, then T=(1=AA')^(-1) is the solution. I+ATA'-T = 0 is the discrete Lyapunov equation, readily solved by for instance matlab (see dlyap.m). dlyap.m converts the discrete Lyapunov equation to a continous Lyapunov equation, using lyap.m for solving it. Lars ============================================================================== [Remark: the series shown does converge if all eigenvalues are less than 1 in magnitude, using the Spectral Radius formula to bound || A^n (A')^n || --djr