Singular matrix

A singular matrix is a square matrix that is not invertible, unlike non-singular matrix which is invertible. Equivalently, an $nxn$ matrix $A$ is singular if and only if determinant, $det(A)=0$. In classical linear algebra, a matrix is called non-singular (or invertible) when it has an inverse; by definition, a matrix that fails this criterion is singular. In more algebraic terms, an $nxn$ matrix A is singular exactly when its columns (and rows) are linearly dependent, so that the linear map $x\rightarrow Ax$ is not one-to-one.

In this case the kernel (null space) of A is non-trivial (has dimension ≥1), and the homogeneous system $Ax=0$ admits non-zero solutions. These characterizations follow from standard rank-nullity and invertibility theorems: for a square matrix A, $det(A)\neq 0$ if and only if $rank(A)=n$, and $det(A)=0$ if and only if $rank(A)<n$.

Key properties and characteristics

• Determinant is zero: By definition an singular matrix have determinant of zero. Consequently, any co-factor expansion or determinant formula yields zero.
• Non-invertible: Since $det(A)=0$, the classic inverse does not exist in the case of singular matrix.
• Rank deficiency: Any structural reason that reduces the rank will cause singularity. For instance, if in a $3x3$ matrix the third row becomes the sum of first two rows, $rank(A)=2>3$ then it is a singular matrix.
• Numerical noise/Round off: In numerical computations, a matrix may be nearly singular when its determinant is extremely small (due to floating-point error or ill-conditioning), effectively causing instability. While not exactly zero in finite precision, such near-singularity can cause algorithms to fail as if singular.

In summary, any condition that forces the determinant to zero or the rank to drop below full automatically yields singularity.

Computational implications of singularity

• No direct inversion: Many algorithms rely on computing A^-1. If $A$ is singular the inversion provides a meaningless value.
• Gaussian-Elimination: In algorithms like Gaussian elimination (LU factorization), encountering a zero pivot signals singularity. In practice, with partial pivoting, the algorithm will fail to find a nonzero pivot in some column if and only if $A$ is singular. Indeed, if no nonzero pivot can be found, the matrix is singular.
• Infinite condition number: The condition number of a matrix (ratio of largest to smallest singular values) is infinite for a truly singular matrix. An infinite condition number means any numerical solution is unstable: arbitrarily small perturbations in data can produce large changes in solutions. In fact, a system is “singular” precisely if its condition number is infinite, and it is “ill-conditioned” if the condition number is very large.
• Information data loss: Geometrically, a singular matrix compresses some dimension(s) to zero (maps whole subspaces to a point or line). In data analysis or modeling, this means information is lost in some directions. For example, in graphics or transformations, a singular transformation (e.g. projection to a line) cannot be reversed.

Applications

• In robotics: In mechanical and robotic systems, singular Jacobian matrices indicate kinematic singularities. For example, the Jacobian of a robotic manipulator (mapping joint velocities to end-effector velocity) loses rank when the robot reaches a configuration with constrained motion. At a singular configuration, the robot cannot move or apply forces in certain directions. This has practical implications for planning and control (avoiding singular poses). Similarly, in structural engineering (finite-element models), a singular stiffness matrix signals an unrestrained mechanism (insufficient boundary conditions), meaning the structure is unstable and can deform without resisting forces.

• Physics and Network theory: In graph theory and network physics, the Laplacian matrix of a graph is inherently singular (it has a zero eigenvalue) because each row sums to zero. This reflects the fact that the uniform vector is in its nullspace. Such singularity encodes fundamental conservation laws (e.g. Kirchhoff’s current law in circuits) or the existence of a connected component. In physics, singular matrices can arise in constrained systems (singular mass or inertia matrices in multibody dynamics, indicating dependent coordinates) or in degenerate Hamiltonians (zero-energy modes).

• Computer science and data analysis: In machine learning and statistics, singular matrices frequently appear due to multicollinearity. For instance, a data matrix $X$ leads to a singular covariance or $X^{T}X$ matrix if features are linearly dependent. This occurs in linear regression when predictors are collinear, causing the normal equations matrix $X^{T}X$ to be singular. The remedy is often to drop or combine features, or use the pseudoinverse. Dimension-reduction techniques like Principal Component Analysis (PCA) exploit SVD: singular value decomposition yields low-rank approximations of data, effectively treating the data covariance as singular by discarding small singular values. In numerical algorithms (e.g. solving linear systems, optimization), detection of singular or nearly-singular matrices signals that specialized methods (pseudo-inverse, regularized solvers) are needed.

• Computer graphics: Certain transformations (e.g. projections from 3D to 2D) are modeled by singular matrices, since they collapse a dimension. Handling these requires care (one cannot invert a projection). In cryptography and coding theory, invertible matrices are used for mixing operations; singular ones would be avoided or detected as errors.

History

The study of singular matrices is rooted in the early history of linear algebra. Determinants were first developed (in Japan by Seki in 1683 and in Europe by Leibniz and Cramer in the 1690s as tools for solving systems of equations. Leibniz explicitly recognized that a system has a solution precisely when a certain determinant expression equals zero. In that sense, singularity (determinant zero) was understood as the critical condition for solvability. Over the 18th and 19th centuries, mathematicians (Laplace, Cauchy, etc.) established many properties of determinants and invertible matrices, formalizing the notion that $det(A)=0$ characterizes non-invertibility.

The term “singular matrix” itself emerged later, but the conceptual importance remained. In the 20th century, generalizations like the Moore–Penrose pseudoinverse were introduced to systematically handle singular or non-square cases. As recent scholarship notes, the idea of a pseudoinverse was proposed by E. H. Moore in 1920 and rediscovered by R. Penrose in 1955, reflecting its longstanding utility. The pseudoinverse and singular value decomposition became fundamental in both theory and applications (e.g. in quantum mechanics, signal processing, and more) for dealing with singularity. Today, singular matrices are a canonical subject in linear algebra: they delineate the boundary between invertible (well-behaved) cases and degenerate (ill-posed) cases. In abstract terms, singular matrices correspond to non-isomorphisms in linear mappings and are thus central to the theory of vector spaces and linear transformations.

Example

Example 1 (2×2 matrix):

$A={\begin{bmatrix}1&2\\2&4\end{bmatrix}}$

Compute its determinant: $det(A)$ = $1.4-2.2=4-4=0$. Thus A is singular. One sees directly that the second row is twice the first, so the rows are linearly dependent. To illustrate failure of invertibility, attempt Gaussian elimination:

• Pivot on the first row; use it to eliminate the entry below:

$Row_{2}\leftarrow Row_{2}-2*Row_{1}$ = ${\begin{bmatrix}1&2\\2-1(2)&4-2(2)\end{bmatrix}}$ = ${\begin{bmatrix}1&2\\0&0\end{bmatrix}}$

Now the second pivot would be the (2,2) entry, but it is zero. Since no nonzero pivot exists in column 2, elimination stops. This confirms $rank(A)=1<2$ and that A has no inverse.

Solving $Ax=b$ exhibits infinite/ no solutions. For example, Ax=0 gives:

$x+2y=0,2x+4y=0$

which are the same equation. Thus the nullspace is one-dimensional, then Ax=b has no solution.