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
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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,
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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]