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Underdetermined system

An underdetermined system of linear equations is a system in which the number of independent equations is fewer than the number of unknowns, typically resulting in either no solution or infinitely many solutions.^[1]^[2] In linear algebra, such systems are commonly expressed in the matrix form \mathbf{Ax} = \mathbf{b}, where \mathbf{A} is an m \times n coefficient matrix with m < n, \mathbf{x} is the n \times 1 vector of unknowns, and \mathbf{b} is the m \times 1 constant vector.^[3] The existence of solutions depends on the rank of \mathbf{A} equaling the rank of the augmented matrix [\mathbf{A} | \mathbf{b}]; if consistent, the solution set is an affine subspace with dimension at least n - m, parameterized by at least n - m free variables.^[3]^[1] To analyze underdetermined systems, Gaussian elimination or row reduction to reduced row echelon form (RREF) is employed, revealing the free variables and expressing dependent variables in terms of them, such as x_1 = c_1 + d_1 r, where r is a free parameter that can take any real value.^[1] The general solution consists of a particular solution plus the null space of \mathbf{A}, which spans the homogeneous solutions.^[3] Geometrically, in \mathbb{R}^3, a consistent underdetermined system with two equations often intersects along a line, representing a continuum of solutions.^[2] Underdetermined systems arise in various applications, including signal processing, optimization, and engineering modeling, where additional constraints or regularization techniques like the Moore-Penrose pseudoinverse may be used to find minimal-norm solutions.^[3] Numerical methods, such as those implemented in libraries like NumPy's linalg.pinv or SciPy's null space computation, facilitate practical solving.^[3]

Fundamentals

Definition

An underdetermined system refers to a system of linear equations given by Ax = b, where A is an m \times n matrix with m < n, indicating fewer equations than unknowns, and x and b are vectors in \mathbb{R}^n and \mathbb{R}^m, respectively (or over the complex numbers \mathbb{C}).^[4]^[5] This structure typically results in either no solutions or infinitely many solutions, depending on the consistency of the system. In contrast, a square system has m = n, potentially yielding a unique solution if A is invertible, while an overdetermined system with m > n often lacks an exact solution unless b lies in the column space of A.^[6] The general form of such a system can be expressed as a set of m linear equations in n variables x_1, x_2, \dots, x_n, where the coefficient matrix A has rank r(A) \leq m < n.^[2] The matrix formulation of linear systems, including underdetermined ones, was advanced in the 19th century through developments in matrix theory, notably by James Joseph Sylvester, who introduced the term 'matrix' in 1850.^[7] A simple illustrative example is the equation x + y = 1, representing one equation in two variables, which admits infinitely many solutions of the form x = t, y = 1 - t for any scalar t.^[8]

Properties and Characteristics

An underdetermined system of linear equations, represented as Ax = b where A is an m \times n matrix with m < n, exhibits several key intrinsic properties stemming from its structure. A fundamental characteristic is the multiplicity of solutions: such systems have either no solution or infinitely many solutions, never a unique one. This arises because the number of equations is insufficient to uniquely determine all variables when the system is consistent. In contrast, overdetermined systems (with m > n) often lack solutions due to overconstraint, though least-squares approximations may be used.^[2] The consistency of the system depends on whether b lies in the column space of A; if it does, the system is consistent and admits solutions, whereas if b is outside the column space, no solutions exist. For consistent underdetermined systems, the solution set is non-unique due to the presence of free variables. The number of these degrees of freedom is given by n - \rank(A), which quantifies the dimensionality of the solution space and leads to infinitely many solutions parameterized by these free variables.^[9]^[10] The rank-nullity theorem provides a precise framework for understanding this non-uniqueness: the nullity of A, or dimension of the null space, equals n - \rank(A), and since m < n implies \nullity(A) > 0 under full row rank, there is a non-trivial null space contributing to multiple solutions. Geometrically, the solution set for a consistent system forms an affine subspace of \mathbb{R}^n with dimension n - \rank(A), translating the null space of A by a particular solution. This affine structure highlights the system's inherent ambiguity, where solutions differ by vectors in the null space.^[11]^[12]

Solutions

General Solution Structure

The general solution to a non-homogeneous underdetermined linear system Ax = b, where A is an m \times n matrix with m < n and the system is consistent, takes the form x = x_p + x_h, with x_p denoting a particular solution satisfying A x_p = b and x_h any vector in the null space of A (i.e., satisfying the homogeneous equation A x_h = 0).^[13] This expression highlights the affine structure of the solution set, which is a translation of the null space by the particular solution.^[14] A particular solution x_p can be found using Gaussian elimination on the augmented matrix [A \mid b], reducing it to row echelon form or reduced row echelon form (RREF).^[15] In the RREF, pivot columns correspond to basic (pivot) variables, while non-pivot columns indicate free variables. To obtain x_p, assign zero to each free variable and back-substitute to solve for the pivot variables.^[13] Alternatively, for a minimum Euclidean norm particular solution (when the rows of A are linearly independent), the Moore-Penrose pseudoinverse provides x_p = A^+ b, where A^+ = A^T (A A^T)^{-1}.^[13] The full parameterization of the solution uses the rank r of A, involving n - r free parameters t_1, \dots, t_{n-r}. Each component of x is then expressed as x_i = (x_p)_i + \sum_{k=1}^{n-r} c_{i k} t_k, where the coefficients c_{i k} derive from a basis for the null space, obtained via the free columns in the RREF of A.^[14] This yields an affine subspace of dimension n - r in \mathbb{R}^n. For illustration, consider the underdetermined system $2x + 3y = 5 in \mathbb{R}^2. Row reduction of the augmented matrix \begin{bmatrix} 2 & 3 & \mid & 5 \end{bmatrix} gives \begin{bmatrix} 2 & 3 & \mid & 5 \end{bmatrix} (already in RREF, with x as pivot variable and y free). Setting y = 0 yields the particular solution x_p = (5/2, 0). The null space basis is found by solving $2x + 3y = 0, giving x = -\frac{3}{2} t, y = t for parameter t. Thus, the general solution is

x = \frac{5}{2} - \frac{3}{2} t, \quad y = t,

parameterizing a line in \mathbb{R}^2.^[15]

Homogeneous Systems

A homogeneous system is a special case of the linear equation Ax = b where the constant vector b = 0, resulting in the equation Ax = 0. Such systems are always consistent, as the zero vector x = 0 satisfies the equation, providing the trivial solution.^[16] The solution space of a homogeneous system Ax = 0 is the null space of the matrix A, denoted N(A), which consists of all vectors x \in \mathbb{R}^n such that Ax = 0. The null space forms a subspace of \mathbb{R}^n. By the rank-nullity theorem, the dimension of N(A), known as the nullity of A, equals n - \rank(A). In underdetermined systems where m < n, \rank(A) \leq m < n, so the nullity is at least n - m > 0, ensuring non-trivial solutions exist./03%3A_The_Fundamental_Subspaces/3.02%3A_Null_Space)^[17] To find a basis for the null space, perform Gaussian elimination (row reduction) on A to obtain its reduced row echelon form. The free variables corresponding to non-pivot columns generate the basis vectors by setting each free variable to 1 while others to 0 and solving for pivot variables. The null space is then spanned by these basis vectors v_1, v_2, \dots, v_k, where k = \nullity(A). The general solution is given by all linear combinations:

x = c_1 v_1 + c_2 v_2 + \dots + c_k v_k, \quad c_i \in \mathbb{R}.

^[18]^[19] The null space exhibits key subspace properties: it is closed under vector addition and scalar multiplication, contains the zero vector, and is non-trivial if its dimension exceeds 0, meaning solutions beyond the trivial one exist./03%3A_The_Fundamental_Subspaces/3.02%3A_Null_Space) For example, consider the underdetermined matrix A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, which is $2 \times 2 with \rank(A) = 1. Row reduction yields the same form, with the second column as a free variable. Setting it to 1 gives the basis vector \begin{pmatrix} 0 \\ 1 \end{pmatrix}, so N(A) = \operatorname{span} \left\{ \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right\}, and solutions are x = c \begin{pmatrix} 0 \\ 1 \end{pmatrix} for c \in \mathbb{R}.^[20]

Extensions

Polynomial Systems

A polynomial system is underdetermined if it consists of m polynomial equations in n variables where m < n, typically considered over the complex numbers \mathbb{C} or real numbers \mathbb{R}.^[21] Such systems arise naturally in algebraic geometry, where the equations define relations among variables without fully specifying unique solutions.^[22] The solution set of an underdetermined polynomial system forms an algebraic variety, the common zero locus of the defining polynomials. By dimension theory in algebraic geometry, this variety has Krull dimension at least n - m, assuming the polynomials generate an ideal of height at most m.^[22] Generically, for systems satisfying suitable independence conditions (such as being a complete intersection), the dimension is exactly n - m > 0, implying a positive-dimensional solution set with infinitely many solutions over algebraically closed fields like \mathbb{C}. This contrasts with the linear case, which is a special instance where solutions parameterize an affine subspace of dimension precisely n - m.^[21] Underdetermination in polynomial systems thus generally yields infinite solutions organized into positive-dimensional components, such as curves or surfaces, rather than isolated points. To analyze or solve these systems, key concepts include ideal membership testing and Gröbner bases, which provide a canonical basis for the ideal generated by the polynomials, enabling elimination of variables and determination of the variety's structure. Gröbner bases facilitate computational approaches to finding solutions or verifying properties like dimension, even in underdetermined cases. For example, consider the underdetermined system of two quadratic equations in three variables:

\begin{align*} x^2 + y^2 &= 1, \\ x^2 + y^2 + z^2 &= 1. \end{align*}

Subtracting the equations yields z^2 = 0, so z = 0, and the first equation describes a circle in the xy-plane, forming a one-dimensional curve (the unit circle in the plane z = 0) as the solution set over \mathbb{R}. This illustrates the typical positive-dimensional outcome, with dimension $3 - 2 = 1.^[21] Unlike linear underdetermined systems, which always produce infinite solutions forming a flat affine space over \mathbb{R} or \mathbb{C}, nonlinear polynomial systems may yield only finite isolated real solutions despite the underdetermination, due to the geometry of real algebraic varieties. For instance, the system

\begin{align*} x^2 + y^2 &= 0, \\ x^2 + y^2 + z^2 &= 1 \end{align*}

has complex dimension 1 (a union of lines), but over \mathbb{R}, the first equation forces x = y = 0, leading to z^2 = 1 and exactly two isolated points: (0, 0, 1) and (0, 0, -1). This phenomenon arises from the nonlinearity restricting the real points on the complex variety.^[21]

Systems with Additional Constraints

Underdetermined linear systems can be augmented with additional linear equality constraints, which effectively increase the number of independent equations and may alter the dimensionality of the solution set. For instance, if the original system has m < n equations where A \in \mathbb{R}^{m \times n} and b \in \mathbb{R}^m, adding k independent equality constraints Cx = d with C \in \mathbb{R}^{k \times n} and d \in \mathbb{R}^k results in an effective system with m + k equations. If m + k = n and the combined coefficient matrix has full rank, the system becomes determined, yielding a unique solution; if m + k > n, it may become overdetermined with potentially no solution or a unique one under consistency.^[23] Inequality constraints, such as nonnegativity x \geq 0, further restrict the solution space without introducing new equations. These constraints define half-spaces in \mathbb{R}^n, and their intersection with the affine solution set of the original equations forms a polyhedron, which is the bounded or unbounded convex set of feasible points. Additional linear inequalities Gx \leq h with G \in \mathbb{R}^{p \times n} and h \in \mathbb{R}^p can reduce the infinite solution set of an underdetermined system to a finite number of points or even render it empty if inconsistent. The resulting polyhedral set is pointed if it contains no line, meaning the lineality space (nullspace of the constraint matrices) is trivial.^[23]^[24] Feasibility of such constrained systems, particularly with inequalities, is characterized by theorems of the alternative like Farkas' lemma. For the system Ax = b, x \geq 0, it states that either there exists a feasible x \geq 0 satisfying the equations, or there exists y \geq 0 such that y^T A \geq 0 and y^T b < 0, but not both. This provides a certificate of infeasibility via a separating hyperplane when no nonnegative solution exists. For general inequalities Ax \leq b, feasibility holds if and only if there is no y \geq 0 with y^T A = 0 and y^T b < 0.^[25]^[26] Consider the underdetermined system x_1 + x_2 = 1 in two variables with the nonnegativity constraint x_1 \geq 0, x_2 \geq 0. Without inequalities, the solution set is the infinite line x_2 = 1 - x_1; with them, it becomes the bounded line segment from (1,0) to (0,1), a polyhedron with vertices at these endpoints. If the right-hand side were -1, the system x_1 + x_2 = -1, x_1 \geq 0, x_2 \geq 0 would be infeasible, as certified by y = 1 > 0 where y (1,1) = (1,1) \geq (0,0) but y (-1) = -1 < 0. Thus, constraints can transform underdetermined systems into ones with finite or empty solution sets, effectively making them determined in terms of bounded feasibility regions.^[25]^[23]

Applications

Optimization Contexts

In linear programming, underdetermined systems occur when the number of decision variables exceeds the number of equality constraints, resulting in feasible regions defined as polyhedra with infinitely many feasible solutions due to the positive dimension of the solution space.^[27] These polyhedra represent the set of points satisfying Ax ≤ b (or Ax = b in equality form), where the underdetermined nature implies that the null space of A has dimension greater than zero, allowing for a continuum of solutions along recession directions.^[28] Optimization over such regions typically seeks to minimize a linear objective function, with optimal solutions attained at vertices of the polyhedron, though the presence of multiple variables often leads to alternative optima spanning edges or faces.^[28] A key application in underdetermined systems is the computation of minimum norm solutions, which select a unique point from the affine solution set by minimizing the Euclidean norm. For a consistent system Ax = b with m < n and full row rank, the solution x = A^+ b, where A^+ denotes the Moore-Penrose pseudoinverse, provides the unique minimizer of |x|_2 subject to Ax = b.^[29]

\mathbf{x} = \mathbf{A}^+ \mathbf{b} = \arg\min_{\mathbf{x}} \|\mathbf{x}\|_2 \quad \text{subject to} \quad \mathbf{Ax} = \mathbf{b}

This pseudoinverse is defined as A^+ = V \Sigma^+ U^T from the SVD A = U \Sigma V^T, where \Sigma^+ inverts the nonzero singular values, ensuring the minimum norm property for underdetermined cases.^[29] Such solutions are particularly useful in regularization contexts, where the minimum norm acts as a proxy for stability in ill-posed problems. To promote sparsity in underdetermined systems, optimization often shifts to L1-norm minimization, which favors solutions with few nonzero entries over the L2 norm. The basis pursuit problem formulates this as minimizing |x|_1 subject to Ax = b, and under conditions like the restricted isometry property (RIP) of order 2s on A, it recovers all exactly s-sparse solutions, where the sparsity level s satisfies s ≲ m / log(n/s).^[30]^[31] This approach links directly to compressed sensing, where underdetermined linear measurements (m << n) of sparse signals are inverted via L1 optimization to enable sub-Nyquist sampling and reconstruction.^[32] In signal processing, basis pursuit exemplifies this: given measurements b = Ax with a sparse underlying signal x (support size k << n) and sensing matrix A of size m × n (m < n), solving \min |x|_1 s.t. Ax = b yields the unique sparsest solution when A satisfies the null space property.^[30] This has high impact in applications like MRI and radar, where sparsity promotion via L1 minimization reduces data requirements while preserving signal fidelity.^[32]

Sparse Solution Techniques

In compressive sensing, underdetermined linear systems A \mathbf{x} = \mathbf{b} with m < n enable the exact recovery of sparse signals \mathbf{x} that have at most s nonzero entries, provided the measurement matrix A satisfies certain conditions; this exploits the fact that sparsity reduces the effective dimensionality, allowing unique reconstruction from fewer measurements than the ambient dimension.^[33] Seminal results show that minimizing the \ell_0-norm (counting nonzeros) or its convex \ell_1-norm proxy recovers the exact sparse solution with high probability for random matrices, assuming s is sufficiently small relative to m.^[33]^[31] Key algorithms for sparse recovery include greedy iterative methods and convex optimization approaches. Orthogonal matching pursuit (OMP) is a greedy algorithm that builds the support of \mathbf{x} iteratively by selecting the column of A most correlated with the current residual \mathbf{r} = \mathbf{b} - A \mathbf{x}_{\text{current}}; specifically, at each step, it chooses the index j = \arg\max_i | \langle \mathbf{a}_i, \mathbf{r} \rangle | / \| \mathbf{a}_i \|_2, where \mathbf{a}_i are the columns of A, then projects the residual orthogonally onto the selected subspace.^[34] Basis pursuit, conversely, formulates recovery as the linear program \min \| \mathbf{x} \|_1 subject to A \mathbf{x} = \mathbf{b}, which promotes sparsity through the \ell_1-norm.^[35] For noisy measurements A \mathbf{x} + \mathbf{e} = \mathbf{b} with \| \mathbf{e} \|_2 \leq \epsilon, basis pursuit denoising extends this to \min \| \mathbf{x} \|_1 subject to \| A \mathbf{x} - \mathbf{b} \|_2 \leq \epsilon, ensuring stable recovery bounds for compressible signals.^[35] A sufficient condition for uniqueness and exact recovery in these methods is the restricted isometry property (RIP) of order $2s, which requires that for all s-sparse \mathbf{h}, (1 - \delta) \| \mathbf{h} \|_2^2 \leq \| A \mathbf{h} \|_2^2 \leq (1 + \delta) \| \mathbf{h} \|_2^2 with \delta < \sqrt{2} - 1; random matrices like Gaussian or partial Fourier satisfy RIP with high probability when m \gtrsim s \log(n/s).^[31] This property ensures that \ell_1-minimization (and greedy methods under similar incoherence) recovers the sparse solution exactly or approximately.^[31] A representative example is recovering a 10-sparse signal in \mathbb{R}^{1000} from 50 linear measurements, where RIP guarantees exact reconstruction via \ell_1-minimization if the matrix is suitably random, demonstrating how underdetermination (50 equations for 1000 unknowns) leverages sparsity for stable inversion.^[31] Overall, underdetermination facilitates sparsity exploitation by allowing algorithms to identify minimal-support solutions amid the infinite general solutions, a capability absent in overdetermined systems.^[33] As a special case, \ell_1-norm optimization underpins many of these techniques.^[35]

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