The Superposition Principle

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Sums of solution to homogeneous systems are also solutions.

Given a matrix equation $A*{\bf x} = {\bf b}$ , its associated homogeneous equation is the equation $A*{\bf x} = {\bf 0}$ (that results from replacing $\bf b$ by $\bf 0$ ). Note that consistency status may change in passing from an arbitrary non-homogeneous equation to its associated counterpart. However, if the original system is consistent, there is an important relation between the two solution sets, which is a manifestation of the superposition principle.

To see this, note that if ${\bf x}', {\bf x}''$ are two solutions to the equation $A*{\bf x} = {\bf b}$ , then $A*({\bf x}' - {\bf x}'') = A*{\bf x}' - A*{\bf x}'' = {\bf b} - {\bf b} = {\bf 0}$ so $({\bf x}' - {\bf x}'')$ is a solution to the associated homogeneous equation. On the other hand, given a solution ${\bf x}_h$ to the associated homogeneous equation, and a solution $\bf x$ to the original equation, we see $A*({\bf x}_h + {\bf x}) = A*{\bf x}_h + A*{\bf x} = {\bf 0} + {\bf b} = {\bf b}$ so ${\bf x}_h + {\bf x}$ is again a solution to the original equation. Thus

Superposition Principle Suppose $A*{\bf x} = {\bf b}$ is a consistent matrix equation, with ${\bf x}_p$ a particular solution to the equation. Then the set of solutions to the original equation can be expressed as $\{{\bf x}_h + {\bf x}_p\ |\ {\bf x}_h\in S_0\}$ where $S_0$ denotes the set of solutions to the associated homogeneous equation.

Suppose we are given a $3\times 5$ matrix $A$ and a $3\times 1$ vector $\bf b$ with $A = \begin {bmatrix} -8 & 8 & 15 & 1 & 11\\ 10 & 11 & 2 & -2 & 13\\ -19 & -7 & -14 & 8& -4 \end {bmatrix},\quad {\bf b} = \begin {bmatrix} 35\\ 5\\ 21 \end {bmatrix}$ and we want to describe the set of solutions to the equation $A*{\bf x} = {\bf b}$ . Using Octave or MATLAB, we compute the rref(ACM): $rref([A\ \ {\bf b}]) = \begin {bmatrix} 1 & 0 & 0 & -475/1419 & -853/4021 & -666/335\\ 0 & 1 & 0 & 636/4021 & 5619/4021 & 9041/4021\\ 0 & 0 & 1 & -789/4021 & -503/4021 & 297/4021 \end {bmatrix}$ The columns (except the right-most) that do not contain leading ones are the fourth and fifth. We also see that the system is consistent. It follows that there is a 2-parameter family of solutions parametrized by the free, or independent variables $x_4$ and $x_5$ , with the dependent variables $x_1, x_2, x_3$ expressable as linear functions in $x_4$ and $x_5$ . Written in standard parametrized form, the full solution set is then given by $\begin{align*} x_1 &= (475/1419)x_4 + (853/4021)x_5 - 666/335 \\ x_2 &= (-636/4021)x_4 - (5619/4021)x_5 + 9041/4021\\ x_3 &= (789/4021)x_4 + (503/4021)x_5 + 297/4021\\ x_4 &= x_4\\ x_5 &= x_5 \end{align*}$

To rewrite in superpositional format, we first choose a particular solution. The easiest one to compute is that corresponding to $x_4 = x_5 = 0$ . In other words the vector determined by the constant terms appearing on the right-hand sides of the above set of equations: ${\bf x}_p = \begin {bmatrix} -666/335\\ 9041/4021\\ 297/4021\\ 0\\ 0 \end {bmatrix}$ The homogeneous part is then given as the 2-parameter set of vectors corresponding to the linear part of the same set of equations: ${\bf x}_h = \begin {bmatrix} (475/1419)x_4 + (853/4021)x_5\\ (-636/4021)x_4 - (5619/4021)x_5\\ (789/4021)x_4 + (503/4021)x_5\\ x_4\\ x_5 \end {bmatrix} = x_4 \begin {bmatrix} 475/1419\\ -636/4021\\ 789/4021\\ 1\\ 0 \end {bmatrix} + x_5 \begin {bmatrix} 853/4021\\ 5619/4021\\ 503/4021\\ 0\\ 1 \end {bmatrix}$ The solution set given above can then be alternately expressed as ${\bf x} = {\bf x}_p + {\bf x}_h$ where ${\bf x}_p$ and ${\bf x}_h$ are as computed above.