Re: Matrix functions
- To: mathgroup at smc.vnet.net
- Subject: [mg42422] Re: Matrix functions
- From: "Richard I. Pelletier" <bitbucket at attbi.com>
- Date: Sun, 6 Jul 2003 06:57:04 -0400 (EDT)
- References: <be5ua2$jhu$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <be5ua2$jhu$1 at smc.vnet.net>, "Dr. Wolfgang Hintze" <weh at snafu.de> wrote: > We can define functions of an nxn matrix M as (infinite) series and > study their convergence using a suitable matrix norm and/or numerically. > .... > Question: Are there standard functions (within some package) of this > kind available? If not, what would be the best way to define e.g. the > exponential matrix function (2), i.e. how to terminate the series > appropriately? there are two std functions available: MatrixExp will compute the exponential of a matrix; MatrixPower will compute a power (e.g. sqrt) of a matrix. for other matrix functions, you have at least three choices other than summing the infinite series. (1) the spectral decomposition theorem, which says that a normal transformation can be wriiten as a linear combination of orthogonal projections, with its eigenvalues as coefficients, and that functions (i.e. power series) of it can be computed by applying those functions just to the eigenvalues. if A = lambda1 E1 + lambda2 E2 then f(A) = f(lambda1) E1 + f(lambda2) E2 (2) since a matrix A satisfies its own characteristic equation, any infinite series in A collapses to a polynomial. that is, if the minimal polynomial of A is of degree 2, then sqrt(A) = aI + bA for some a and b. (solving for a and b is easily done since the eigenvalues of A must also satisfy an eq of that form. ok, if there are repeated eigenvalues, we have to match derivatives too.) (3) easiest of all is that if the matrix can be diagonalized (D = Pinv.A.P) then f(A) = P f(D) pinv. i hope this outline helps you find more detail. rip -- NOTE: email address is rip1 AT comcast DOT net