Regular matrix

In mathematics, the term regular matrix has several meanings, primarily in linear algebra. These include non-singular square matrices, regular matrix pencils, and regular stochastic matrices, as detailed in the following sections. One common usage refers to a square matrix over a field (such as the real or complex numbers) whose determinant is nonzero, equivalently, one that possesses a multiplicative inverse.^[1] This property distinguishes regular matrices from singular matrices, which have zero determinant and no inverse. In the context of linear algebra, a regular matrix represents a bijective linear transformation between vector spaces of equal dimension, ensuring that systems of linear equations involving such a matrix have unique solutions. Key properties of regular matrices include full row and column rank equal to the matrix dimension, allowing for unique solvability of the equation Ax = b for any vector b. The inverse of a regular matrix A, denoted A^{-1}, satisfies A A^{-1} = A^{-1} A = I, where I is the identity matrix, and can be computed using methods such as Gaussian elimination, LU decomposition, or the adjugate formula A^{-1} = \frac{1}{\det(A)} \adj(A). Regular matrices are closed under multiplication and inversion, forming the general linear group GL(n, \mathbb{F}) over a field \mathbb{F}.^[1] In applied contexts, such as numerical analysis and control theory, regularity ensures numerical stability in algorithms like solving linear systems or eigenvalue computations, though partial pivoting may be required to avoid instability even for regular matrices. Over rings rather than fields, the notion extends to von Neumann regular matrices, where a matrix A admits a generalized inverse B such that ABA = A, but this is distinct from the classical linear algebra definition.^[2]

Mathematics

Non-singular square matrices

In linear algebra, a regular matrix is defined as a square matrix A \in \mathbb{R}^{n \times n} (or more generally over a field) that is invertible, meaning there exists another square matrix B of the same size such that AB = BA = I_n, where I_n is the n \times n identity matrix.^[3] The matrix B is called the inverse of A and is denoted A^{-1}.^[4] This definition equates the term "regular" with "nonsingular" or "invertible," distinguishing it from singular matrices, which lack an inverse.^[3] Several conditions are equivalent to a square matrix A being regular. These include: the determinant \det(A) \neq 0; the rank of A equals n (full rank); the columns (or rows) of A are linearly independent; the image of A spans the entire space \mathbb{R}^n; the kernel of A is only the zero vector; and for every vector b \in \mathbb{R}^n, the linear system Ax = b has a unique solution x = A^{-1}b.^[3]^[5] Regular matrices possess several fundamental properties. The inverse A^{-1} is unique for a given regular A.^[3] If A is regular, then so is A^{-1}, and (A^{-1})^{-1} = A.^[4] The product of two regular matrices is regular, with (AB)^{-1} = B^{-1}A^{-1}.^[3] Additionally, if A is regular, its transpose A^T is regular, and (A^T)^{-1} = (A^{-1})^T.^[4] The inverse can also be expressed using the adjugate matrix as A^{-1} = \frac{1}{\det(A)} \adj(A), where \adj(A) is the transpose of the cofactor matrix of A.^[6] To determine if a matrix is regular and compute its inverse, Gaussian elimination (or row reduction) is a standard method. Form the augmented matrix [A \mid I_n] and perform row operations to transform the left part to the identity matrix I_n; if successful, the right part becomes A^{-1}, confirming A is regular.^[4] If the left part cannot be reduced to I_n, A is singular. For a $2 \times 2 matrix A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} with \det(A) = ad - bc \neq 0, the inverse is explicitly given by

A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}.

^[7] This formula follows directly from the adjugate expression.^[6] The term "regular matrix" as a synonym for nonsingular appears in older linear algebra texts from the 19th and 20th centuries, though it is less common in modern English usage compared to "invertible" or "nonsingular."^[4]

Regular matrix pencils

In linear algebra, a matrix pencil consists of a pair of n \times n matrices A and B over a field, typically the complex numbers, and is expressed as the matrix-valued polynomial \lambda B - A. A matrix pencil is regular if the determinant \det(\lambda B - A) is not the zero polynomial, meaning it is a polynomial of degree exactly n.^[8]^[9] Equivalent conditions for regularity include the pencil possessing exactly n eigenvalues, counting multiplicities, which may be finite or infinite; alternatively, not all n \times n minors of \lambda B - A vanish identically as polynomials.^[10]^[11] This contrasts with singular pencils, where \det(\lambda B - A) \equiv 0, implying fewer than n eigenvalues. A key property of regular pencils is their transformation into Weierstrass canonical form via equivalence transformations, where nonsingular matrices P and Q satisfy P(\lambda B - A)Q = \operatorname{diag}(J(\lambda), N(\lambda)). Here, J(\lambda) is a Jordan-like block diagonal matrix capturing finite eigenvalues, while N(\lambda) consists of nilpotent Jordan blocks for infinite eigenvalues, with the sizes reflecting the algebraic and geometric multiplicities.^[10]^[8] Regular matrix pencils arise in applications such as control theory, where the generalized eigenvalue problem \det(\lambda B - A) = 0 informs pole placement and system stability analysis for descriptor systems of the form B \dot{x} = A x.^[11] Additionally, low-rank perturbations of a regular pencil preserve regularity under conditions where the perturbation does not increase the nullity beyond the original structure, allowing controlled changes to the eigenvalue spectrum while maintaining the Weierstrass form's integrity.^[12] For example, consider the $2 \times 2 matrices A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} and B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}; then \lambda B - A = \begin{pmatrix} \lambda & -1 \\ 0 & \lambda \end{pmatrix}, so \det(\lambda B - A) = \lambda^2, a degree-2 polynomial confirming regularity with a double eigenvalue at \lambda = 0. In contrast, for A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} and B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \lambda B - A = \begin{pmatrix} -1 & \lambda \\ 0 & 0 \end{pmatrix}, yielding \det(\lambda B - A) = 0 identically, marking a singular pencil.^[8]^[13]

Regular stochastic matrices

A regular stochastic matrix, also known as a primitive stochastic matrix in the context of non-negative matrix theory, is a square matrix P with non-negative entries where each row sums to 1, and there exists some positive integer k such that P^k has all strictly positive entries.^[14] This property ensures that the associated Markov chain mixes thoroughly over iterations, avoiding persistent zero probabilities in transitions.^[15] Equivalent conditions for a stochastic matrix to be regular include the underlying directed graph being strongly connected (corresponding to irreducibility of the Markov chain) and the greatest common divisor of all cycle lengths being 1 (corresponding to aperiodicity).^[16] These conditions guarantee that the matrix is primitive, meaning its powers eventually become positive, as established in the theory of non-negative matrices.^[16] Key properties of regular stochastic matrices include the convergence of their powers P^n as n \to \infty to a limiting matrix where all rows are identical and equal to the unique stationary probability vector \pi, satisfying \pi P = \pi and \sum_i \pi_i = 1.^[16] This stationary vector represents the long-term proportion of time spent in each state, independent of the initial distribution. The rate of this convergence is governed by the modulus of the second-largest eigenvalue \lambda_2 of P, where |\lambda_2| < 1, with faster mixing occurring when |\lambda_2| is smaller.^[17] Regular stochastic matrices are non-singular, inheriting invertibility from their stochastic structure while imposing non-negativity constraints not present in general square matrices. In applications to Markov chains, regular stochastic matrices ensure ergodicity, meaning the chain converges to a unique limiting distribution from any starting state, which is crucial for modeling systems with eventual equilibrium such as population dynamics or queueing processes.^[16] Variants appear in the PageRank algorithm, where a damping factor modifies the web's hyperlink graph into a regular stochastic matrix to compute stable importance scores for web pages via the stationary distribution.^[18] For example, consider the 2×2 stochastic matrix

P = \begin{pmatrix} 0.9 & 0.1 \\ 0.2 & 0.8 \end{pmatrix}.

Computing P^2 yields

P^2 = \begin{pmatrix} 0.83 & 0.17 \\ 0.34 & 0.66 \end{pmatrix},

where all entries are positive, confirming regularity.^[19] In contrast, the periodic matrix

Q = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

satisfies Q^2 = I, and no power of Q has all positive entries, illustrating the failure of regularity due to aperiodicity.^[20]

Other uses

Quadraphonic audio encoding

The Regular Matrix (RM) encoding, also known as the QS (Quadraphonic Source) system, was developed by the Japanese electronics company Sansui and introduced commercially in 1971 as a method to deliver four-channel surround sound over standard two-channel stereo media.^[21] This 2x2 matrix technique encodes front left (FL), front right (FR), rear left (RL), and rear right (RR) audio channels into left (L) and right (R) stereo signals for transmission via vinyl records, FM radio, or tape, enabling backward compatibility with existing stereo equipment.^[21] The system was offered royalty-free to record labels, leading to its adoption by smaller producers and resulting in hundreds of QS-encoded LP releases during the 1970s, including titles from labels like Vox, Turnabout, and ABC.^[21] In the encoding process, the channels are combined with amplitude adjustments and 90-degree phase shifts for the rear channels to minimize interference; for example, the left channel is formed as L = 0.924 FL + 0.383 FR + 0.924 RL i + 0.383 RR i, and the right channel as R = 0.383 FL + 0.924 FR - 0.383 RL i - 0.924 RR i (where i represents the 90-degree phase shift).^[22] Decoding occurs through matrix circuits in compatible receivers, which apply phase detection and steering logic (such as Sansui's Vario-Matrix) to approximate the original four channels, achieving separations of 3-10 dB in early models and up to 35-40 dB in later designs.^[21] This approach ensured bandwidth efficiency by fitting quadraphonic audio within the standard stereo spectrum but introduced some channel crosstalk, particularly in rear-to-front isolation, as an inherent trade-off of the analog matrix design.^[22] Key features of RM encoding included full compatibility with mono and stereo playback—where rear signals would largely cancel out, preserving the front stereo image—and support for ambient surround effects without requiring discrete channels.^[21] In contrast to the competing SQ system developed by Sony and CBS, which relied on 90-degree phase shifts primarily for rear channels with fixed amplitudes, QS emphasized balanced front-rear encoding for improved diagonal imaging and better decoding of non-QS matrix sources.^[22] The technology powered dedicated QS receivers and adapters from manufacturers like Sansui, Pioneer, and Marantz until the decline of analog quadraphonics in the late 1970s, driven by format incompatibilities and the rise of digital audio.^[21] Although largely supplanted by modern digital surround formats like Dolby Digital and DTS, which offer discrete multi-channel encoding without crosstalk limitations, the Regular Matrix system has seen revived interest among audio enthusiasts and historians for its role in pioneering consumer surround sound, with software emulations and archival reissues preserving QS recordings for contemporary playback.^[21]