Solving symbolic linear equations with maple
How can I solve linear equations of the following type in Maple?
$$\begin{pmatrix} 1 & 1 & 1 & 1\\ b-c & c-b & a-b &0 \\ b-d & d-a & 0 &a-b \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \\ x_3\\ x_4 \end{pmatrix} =\begin{pmatrix} 0\\ 0 \\ 0 \end{pmatrix}$$
The problem is that I want to use $a,b,c,d$ as symbolic variables. It seems, Maple only can solve such an equation when $a,b,c,d$ are fixed numbers.
$\endgroup$ 01 Answer
$\begingroup$I don't see Maple having any problem with generating a symbolic solution to this under-determined system, using either the LinearSolve or the solve commands.
restart:
M := Matrix( [[1, 1, 1, 1], [b-c, c-b, a-b, 0], [b-d, d-a, 0, a-b]] ); [ 1 1 1 1 ] [ ] M := [b - c c - b a - b 0 ] [ ] [b - d d - a 0 a - b]
B := Vector([0, 0, 0]); [0] [ ] B := [0] [ ] [0]Unsurprisingly we get a solution with one free parameter.
X := LinearAlgebra:-LinearSolve(M, B); [ _t[3] (2 a - c - d) ] [ - ------------------- ] [ 3 (b - c) ] [ ] [_t[3] (a - 3 b + c + d)] [-----------------------] X := [ 3 (b - c) ] [ ] [ _t[3] ] [ ] [ (a + c - 2 d) _t[3] ] [ ------------------- ] [ 3 (b - c) ]
map( simplify, M . X - B ); [0] [ ] [0] [ ] [0]The above is all in Matrix-Vector form. Equation form can also be produced straightforwardly.
Xsymb := Vector([x1,x2,x3,x4]); [x1] [ ] [x2] Xsymb := [ ] [x3] [ ] [x4]
ans := Equate( Xsymb, X ); [ _t[3] (2 a - c - d) _t[3] (a - 3 b + c + d) ans := [x1 = - -------------------, x2 = -----------------------, x3 = _t[3], [ 3 (b - c) 3 (b - c) (a + c - 2 d) _t[3]] x4 = -------------------] 3 (b - c) ]Or, we could reformulate the question in terms of explicit equations, and pass those to the solve command.
LinearAlgebra:-GenerateEquations( Matrix([M,B]), convert(Xsymb,list) ); [x1 + x2 + x3 + x4 = 0, a x3 + b x1 - b x2 - b x3 - c x1 + c x2 = 0, -a x2 + a x4 + b x1 - b x4 - d x1 + d x2 = 0]
solve( %, [x1, x2, x4] ); [[ x3 (2 a - c - d) x3 (a - 3 b + c + d) x3 (a + c - 2 d)]] [[x1 = - ----------------, x2 = --------------------, x4 = ----------------]] [[ 3 (b - c) 3 (b - c) 3 (b - c) ]]More in general
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