nLab inverse matrix

Idea

$n\times n$ matrices with entries in a unital ring $R$ form a unital ring $Mat_{n \times n}(R)$ with unit $I_n$ (diagonal matrix whose each entry at the main diagonal is $1_R$). A matrix $A\in Mat_{n \times n}(R)$ is invertible (also said regular) if it has a two-sided inverse $A^{-1}$ in that unital ring, called the matrix inverse (or inverse matrix) of $A$. In other words, it is a matrix satisfying $A A^{-1} = I_n = A^{-1} A$.

Variants

One-sided inverses

Sometimes one-sided inverses in $Mat_{n \times n}(R)$ are also useful as well as one-sided inverses of rectangular matrices.

Schur complement

Sometimes, an inverse matrix does not exist but formulas for some entries of inverse matrix make sense. Related expressions include quasideterminants and Schur complement, see there.

Moore–Penrose inverse

A generalized inverse $A^{-1}$ satisfying weaker requirement $A A^{-1} A = I_n$ and $A^{-1} = A^{-1} A A^{-1}$ is sometimes useful, especially in applied mathematics and approximation theory. These identities play role also in inverse semigroups.

In the case of matrices, such generalized inverse is known as Moore–Penrose inverse.

Properties

Proposition

(fundamental theorem of invertible matrices)

If $R = k$ is a field, $n \in \mathbb{N}$ and $A \in Mat_{n \times n}(k)$ a square matrix, the following are equivalent:

1. $A$ is an invertible matrix;

2. $A$ is the matrix product of elementary matrices.

Related concepts

• quasideterminant

• Gaussian elimination

• rank of a matrix

• matrix group

• general linear group

• matrix equivalence

References

• Wikipedia, Invertible matrix, Moore–Penrose inverse

• WolframMathWorld, Invertible Matrix Theorem

Formulas for inverses of block matrices see shortly at Schur complement and more at

• D. Krob, B. Leclerc, Sec 2 in: Minor identities for quasi-determinants and quantum determinants, Comm. Math. Phys. 169 (1995) 1-23 [doi:10.1007/BF02101594, arXiv:hep-th/9411194]đ

• Chapter 13 (e.g. 13.8), Block LU factorization, in: Nicholas J. Higham, Accuracy and stability of numerical algorithms, Society for Industrial and Applied Mathematics, Year: 2002

• Tzon-Tzer Lu, Sheng-Hua Shiou, Inverses of $2 \times 2$ block matrices, Computers & Mathematics with Applications 43 1–2 (2002) 119-129 [doi:10.1016/S0898-1221(01)00278-4]

• Müge Saadetoğlu, Şakir Mehmet Dinsev, Inverses and determinants of $n \times n$ block matrices, Mathematics 11 17 (2023) 3784 [doi:10.3390/math11173784]