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Question: How to check the consistency of a system of equations (and solve the unknowns) using the LinearAlgebra package

Question:How to check the consistency of a system of equations (and solve the unknowns) using the LinearAlgebra package

Harry Garst 239 Maple
linearalgebra
March 28 2006
0
I am working on a Maximum Likelihood problem and would like to know whether the LinearAlgebra package can be helpful. Up to now I have been using Maple only to check my work. However, someone pointed out that Maple's capabilities reach much further: it also can determine whether the system of equations is consistent and, if so, Maple can solve the equations. Unfortunately, I am only trained as a psychologist and I am still at the beginning of learning Maple's capabilities. Here are the three equations: f = (Y'b - 1t'b)(b'b)^-1 equation#1 where f is a q by 1 vector of unknown constants, Y is a n by q matrix of known constants and Y' is the transpose of Y b is a n by 1 vector of unknown constants, 1 is a q by 1 unit vector (all values are 1), ^-1 denotes the inverse t = Y1(1'1)^-1 or (1/q)Y1 equation#2 where t is a n by 1 vector of unknowns and q is a known scalar (First I found the equations with normal algebra and t is a vector with the row means of Y as its values, therefore (1'1)^-1 is equal to 1/q) b = (1/q)Yb equation#3 In addition, the following constraints hold: f'f= q and f'1 = 0 where O is a scalar (Actually, it is a longitudinal factor analysis problem and b represents the vector of factor loadings, t represents the vector of item intercepts and f is the vector of factor scores. To set the scale of f the mean of f is fixed to zero and its variance is fixed to 1. The number of items is denoted by n and the number of measurement occasions is denoted by q. As I said, I first used normal algebra to differentiate the ML and (b'b)^-1 is a scalar). Can I test using Maple whether this system of equations is consistent and, if so, can Maple solve the unknowns vectors? It would be great to show the capabilities of Maple, because I know that there are quite a few statistically oriented psychologists who are using Maple. Thanks in advance. Harry Garst
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