Draft:Singular matrix

A singular matrix is a square matrix that is not invertible. Equivalently, an $nxn$ matrix $A$ is singular if and only if$det(A)=0$. In classical linear algebra, a matrix is called nonsingular (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, $det(A)=0$ 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 Physics, Engineering, and Computer Science

• 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.