Do the chain rules for matrix functions hold without commutativity?
نویسندگان
چکیده
This paper shows that the commutativity condition [A(t), A′(t)] = 0 is often not necessary to guarantee the chain rules for matrix functions: d dtf(A(t)) = A ′(t)f ′(A(t)) and d dtf(A(t)) = f ′(A(t))A′(t), where A(t) is a square matrix of differentiable functions and f is an analytic function. A further question on the chain rules is presented and discussed.
منابع مشابه
A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential
It is known that the solution to a Cauchy problem of linear differential equations: x′(t) = A(t)x(t), with x(t0) = x0, can be presented by the matrix exponential as exp( ∫ t t0 A(s) ds)x0, if the commutativity condition for the coefficient matrix A(t) holds: [ ∫ t t0 A(s) ds, A(t) ] = 0. A natural question is whether this is true without the commutativity condition. To give a definite answer to...
متن کاملCoordinate Selection Rules for Gibbs Sampling
This paper studies several di erent plans for selecting coordinates for updating via Gibbs sampling. It exploits the inherent features of the Gibbs sampling formulation, most notably its neighborhood structure, to characterize and compare the plans with regard to convergence to equilibrium and variance of the sample mean. Some of the plans rely on completely or almost completely random coordina...
متن کاملRules for Numbers
The real numbers are governed by a collection of rules that have to do with addition, multiplication, and inequalities. In the rules below, x, y, z 2 R. (In other words, x, y, and z are real numbers.) Rules of addition. • (x + y) + z = x + (y + z) (Law of associativity) • x + y = y + x (Law of commutativity) • x + 0 = x (Law of identity) • x + x = 0 (Law of inverses) Rules of multiplication. • ...
متن کاملNew Shewhart-type synthetic bar{X} control schemes for non-normal data
In this paper, Burr-type XII ̄X synthetic schemes are proposed as an alternative to the classical ̄X synthetic schemes when the assumption of normality fails to hold. First, the basic design of the Burr-type XII ̄X synthetic scheme is developed and its performance investigated using exact formulae. Secondly, the non-side-sensitive and side-sensitive Burr-type XII ̄X synthetic schemes are int...
متن کاملEvolutionary games with variable payoffs.
Matrix games, defined by a set of strategies and a corresponding matrix of payoffs, are commonly used to model animal populations because they are both simple and generate meaningful results. It is generally assumed that payoffs are independent of time. However, the timing of contests in real populations may have a marked effect on the value of rewards. We consider matrix games where the payoff...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009