TheInfoList

OR:

In linear algebra, an -by-
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :$\mathbf = \mathbf = \mathbf_n \$ where denotes the -by-
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
and the multiplication used is ordinary
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicond ...
its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If is -by- and the rank of is equal to (), then has a left inverse, an -by- matrix such that . If has rank (), then it has a right inverse, an -by- matrix such that . While the most common case is that of matrices over the real or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
numbers, all these definitions can be given for matrices over any ring. However, in the case of the ring being commutative, the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than being nonzero. For a noncommutative ring, the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings. The set of invertible matrices together with the operation of matrix multiplication (and entries from ring ) form a group, the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
of degree , denoted .

# Properties

## The invertible matrix theorem

Let be a square -by- matrix over a field (e.g., the field of real numbers). The following statements are equivalent (i.e., they are either all true or all false for any given matrix): * There is an -by- matrix such that . * The matrix has a left inverse (that is, there exists a such that ) ''or'' a right inverse (that is, there exists a such that ), in which case both left and right inverses exist and . * is invertible, that is, has an inverse, is nonsingular, and is nondegenerate. * is row-equivalent to the -by-
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
. * is column-equivalent to the -by-
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
. * has pivot positions. * has full rank; that is, . * Based on the rank , the equation has only the trivial solution and the equation has exactly one solution for each in . * The kernel of is trivial, that is, it contains only the null vector as an element, * The columns of are
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ar ...
. * The columns of span . * . * The columns of form a
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting o ...
of . * The linear transformation mapping to is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
from to . * The determinant of is nonzero: .In general, a square matrix over a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
is invertible if and only if its determinant is a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (al ...
in that ring. * The number 0 is not an
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of . * The
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The t ...
is an invertible matrix (hence rows of are
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ar ...
, span , and form a
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting o ...
of ). * The matrix can be expressed as a finite product of elementary matrices.

## Other properties

Furthermore, the following properties hold for an invertible matrix : * $\left(\mathbf A^\right)^ = \mathbf A$ * $\left(k \mathbf A\right)^ = k^ \mathbf A^$ for nonzero scalar * $\left(\mathbf\right)^+ = \mathbf x^+ \mathbf A^$ if has orthonormal columns, where denotes the
Moore–Penrose inverse In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Roge ...
and is a vector * $\left(\mathbf A^\top\right)^ = \left(\mathbf A^\right)^\top$ * For any invertible -by- matrices and , $\left(\mathbf\right)^ = \mathbf B^ \mathbf A^.$ More generally, if $\mathbf A_1, \dots, \mathbf A_k$ are invertible -by- matrices, then $\left(\mathbf A_1 \mathbf A_2 \cdots \mathbf A_ \mathbf A_k\right)^ = \mathbf A_k^ \mathbf A_^ \cdots \mathbf A_2^ \mathbf A_1^.$ *$\det \mathbf A^ = \left(\det \mathbf A\right)^.$ The rows of the inverse matrix of a matrix are
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...
to the columns of (and vice versa interchanging rows for columns). To see this, suppose that where the rows of are denoted as $v_i^$ and the columns of as $u_j$ for $1 \leq i,j \leq n.$ Then clearly, the Euclidean inner product of any two $v_i^ u_j = \delta_.$ This property can also be useful in constructing the inverse of a square matrix in some instances, where a set of
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
vectors (but not necessarily orthonormal vectors) to the columns of are known. In which case, one can apply the iterative Gram–Schmidt process to this initial set to determine the rows of the inverse . A matrix that is its own inverse (i.e., a matrix such that and ), is called an involutory matrix.

## In relation to its adjugate

The adjugate of a matrix can be used to find the inverse of as follows: If is an invertible matrix, then : $A^ = \frac \operatorname\left(A\right).$

## In relation to the identity matrix

It follows from the associativity of matrix multiplication that if : $\mathbf = \mathbf \$ for ''finite square'' matrices and , then also : $\mathbf = \mathbf\$

## Density

Over the field of real numbers, the set of singular -by- matrices, considered as a subset of is a null set, that is, has Lebesgue
measure zero In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null ...
. This is true because singular matrices are the roots of the determinant function. This is a continuous function because it is a polynomial in the entries of the matrix. Thus in the language of
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
,
almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathem ...
-by- matrices are invertible. Furthermore, the -by- invertible matrices are a
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
open set in the
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
of all -by- matrices. Equivalently, the set of singular matrices is closed and nowhere dense in the space of -by- matrices. In practice however, one may encounter non-invertible matrices. And in numerical calculations, matrices which are invertible, but close to a non-invertible matrix, can still be problematic; such matrices are said to be
ill-conditioned In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the inpu ...
.

# Examples

An example with rank of n-1 to be a non-invertible matrix : $\mathbf = \begin 2 & 4\\ 2 & 4 \end .$ We can easily see the rank of this 2*2 matrix is one, which is n-1≠n, so it is a non-invertible matrix. Consider the following 2-by-2 matrix: : $\mathbf = \begin-1 & \tfrac \\ 1 & -1\end .$ The matrix $\mathbf$ is invertible. To check this, one can compute that $\det \mathbf = -\frac$, which is non-zero. As an example of a non-invertible, or singular, matrix, consider the matrix : $\mathbf = \begin -1 & \tfrac \\ \tfrac & -1 \end .$ The determinant of $\mathbf$ is 0, which is a necessary and sufficient condition for a matrix to be non-invertible.

# Methods of matrix inversion

## Gaussian elimination

Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
is a useful and easy way to compute the inverse of a matrix. To compute a matrix inverse using this method, an augmented matrix is first created with the left side being the matrix to invert and the right side being the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
. Then, Gaussian elimination is used to convert the left side into the identity matrix, which causes the right side to become the inverse of the input matrix. For example, take the following matrix: $\mathbf = \begin-1 & \tfrac \\ 1 & -1\end.$ The first step to compute its inverse is to create the augmented matrix $\left\left(\begin -1 & \tfrac & 1 & 0 \\ 1 & -1 & 0 & 1 \end\right\right) .$ Call the first row of this matrix $R_1$ and the second row $R_2$. Then, add row 1 to row 2 $\left(R_1 + R_2 \to R_2\right).$ This yields $\left\left(\begin -1 & \tfrac & 1 & 0 \\ 0 & \tfrac & 1 & 1 \end\right\right).$ Next, subtract row 2, multiplied by 3, from row 1 $\left(R_1 - 3\, R_2 \to R_1\right),$ which yields $\left\left(\begin -1 & 0 & -2 & -3 \\ 0 & \tfrac & 1 & 1 \end\right\right).$ Finally, multiply row 1 by –1 $\left(-R_1 \to R_1\right)$ and row 2 by 2 $\left(2\, R_2 \to R_2\right).$ This yields the identity matrix on the left side and the inverse matrix on the right:$\left\left(\begin 1 & 0 & 2 & 3 \\ 0 & 1 & 2 & 2 \end\right\right).$ Thus, $\mathbf^ = \begin 2 & 3 \\ 2 & 2 \end.$ The reason it works is that the process of Gaussian Elimination can be viewed as a sequence of applying left matrix mutliplication using elementary row operations using
Elementary matrix In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multipli ...
($\mathbf E_n$), such as $\mathbf E_n \mathbf E_ \cdots \mathbf E_2 \mathbf E_1 \mathbf A = \mathbf I.$ Applying right-multiplication using $\mathbf A^,$ we get $\mathbf E_n \mathbf E_ \cdots \mathbf E_2 \mathbf E_1 \mathbf I = \mathbf I \mathbf A^.$ And the right side $\mathbf I \mathbf A^ = \mathbf A^,$ which is the inverse we want. To obtain $\mathbf E_n \mathbf E_ \cdots \mathbf E_2 \mathbf E_1 \mathbf I,$ we create the augumented matrix by combining with and applying
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
. The two portions will be transformed using the same sequence of elementary row operations. When the left portion becomes , the right portion applied the same elementary row operation sequence will become .

## Newton's method

A generalization of
Newton's method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real- ...
as used for a multiplicative inverse algorithm may be convenient, if it is convenient to find a suitable starting seed: :$X_ = 2X_k - X_k A X_k.$ Victor Pan and John Reif have done work that includes ways of generating a starting seed. Byte magazine summarised one of their approaches. Newton's method is particularly useful when dealing with families of related matrices that behave enough like the sequence manufactured for the homotopy above: sometimes a good starting point for refining an approximation for the new inverse can be the already obtained inverse of a previous matrix that nearly matches the current matrix, for example, the pair of sequences of inverse matrices used in obtaining matrix square roots by Denman–Beavers iteration; this may need more than one pass of the iteration at each new matrix, if they are not close enough together for just one to be enough. Newton's method is also useful for "touch up" corrections to the Gauss–Jordan algorithm which has been contaminated by small errors due to imperfect computer arithmetic.

## Cayley–Hamilton method

The Cayley–Hamilton theorem allows the inverse of to be expressed in terms of , traces and powers of : : $\mathbf^ = \frac \sum_^ \mathbf^s \sum_ \prod_^ \frac \operatorname\left\left(\mathbf^l\right\right)^,$ where is dimension of , and is the trace of matrix given by the sum of the main diagonal. The sum is taken over and the sets of all $k_l \geq 0$ satisfying the linear
Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...
: $s + \sum_^ lk_l = n - 1.$ The formula can be rewritten in terms of complete Bell polynomials of arguments $t_l = - \left(l - 1\right)! \operatorname\left\left(A^l\right\right)$ as : $\mathbf^ = \frac \sum_^n \mathbf^ \frac B_\left(t_1, t_2, \ldots, t_\right).$

## Eigendecomposition

If matrix can be eigendecomposed, and if none of its eigenvalues are zero, then is invertible and its inverse is given by : $\mathbf^ = \mathbf\mathbf^\mathbf^,$ where is the square matrix whose -th column is the eigenvector $q_i$ of , and is the
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ...
whose diagonal elements are the corresponding eigenvalues, that is, $\Lambda_ = \lambda_i.$ If is symmetric, is guaranteed to be an orthogonal matrix, therefore $\mathbf^ = \mathbf^\top .$ Furthermore, because is a diagonal matrix, its inverse is easy to calculate: :

## Cholesky decomposition

If matrix is positive definite, then its inverse can be obtained as : $\mathbf^ = \left\left(\mathbf^*\right\right)^ \mathbf^ ,$ where is the lower triangular
Cholesky decomposition In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for ef ...
of , and denotes the conjugate transpose of .

## Analytic solution

Writing the transpose of the matrix of cofactors, known as an adjugate matrix, can also be an efficient way to calculate the inverse of ''small'' matrices, but this recursive method is inefficient for large matrices. To determine the inverse, we calculate a matrix of cofactors: : $\mathbf^ = \mathbf^\mathrm = \begin \mathbf_ & \mathbf_ & \cdots & \mathbf_ \\ \mathbf_ & \mathbf_ & \cdots & \mathbf_ \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf_ & \mathbf_ & \cdots & \mathbf_ \\ \end$ so that : $\left\left(\mathbf^\right\right)_ = \left\left(\mathbf^\right\right)_ = \left\left(\mathbf_\right\right)$ where is the determinant of , is the matrix of cofactors, and represents the matrix
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The t ...
.

### Inversion of 2 × 2 matrices

The ''cofactor equation'' listed above yields the following result for matrices. Inversion of these matrices can be done as follows: : $\mathbf^ = \begin a & b \\ c & d \\ \end^ = \frac \begin \,\,\,d & \!\!-b \\ -c & \,a \\ \end = \frac \begin \,\,\,d & \!\!-b \\ -c & \,a \\ \end.$ This is possible because is the reciprocal of the determinant of the matrix in question, and the same strategy could be used for other matrix sizes. The Cayley–Hamilton method gives :

### Inversion of 3 × 3 matrices

A computationally efficient matrix inversion is given by : $\mathbf^ = \begin a & b & c\\ d & e & f \\ g & h & i\\ \end^ = \frac \begin \, A & \, B & \,C \\ \, D & \, E & \, F \\ \, G & \, H & \, I\\ \end^\mathrm = \frac \begin \, A & \, D & \,G \\ \, B & \, E & \,H \\ \, C & \,F & \, I\\ \end$ (where the scalar is not to be confused with the matrix ). If the determinant is non-zero, the matrix is invertible, with the elements of the intermediary matrix on the right side above given by : $\begin A &=& \left(ei - fh\right), &\quad& D &=& -\left(bi - ch\right), &\quad& G &=& \left(bf - ce\right), \\ B &=& -\left(di - fg\right), &\quad& E &=& \left(ai - cg\right), &\quad& H &=& -\left(af - cd\right), \\ C &=& \left(dh - eg\right), &\quad& F &=& -\left(ah - bg\right), &\quad& I &=& \left(ae - bd\right). \\ \end$ The determinant of can be computed by applying the rule of Sarrus as follows: : $\det\left(\mathbf\right) = aA + bB + cC.$ The Cayley–Hamilton decomposition gives : The general inverse can be expressed concisely in terms of the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
and
triple product In geometry and algebra, the triple product is a product of three 3- dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector ...
. If a matrix $\mathbf = \begin \mathbf_0 & \mathbf_1 & \mathbf_2\end$ (consisting of three column vectors, $\mathbf_0$, $\mathbf_1$, and $\mathbf_2$) is invertible, its inverse is given by : $\mathbf^ = \frac\begin ^\mathrm \\ ^\mathrm \\ ^\mathrm \end.$ The determinant of , , is equal to the triple product of , , and —the volume of the
parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean ...
formed by the rows or columns: : $\det\left(\mathbf\right) = \mathbf_0\cdot\left(\mathbf_1\times\mathbf_2\right).$ The correctness of the formula can be checked by using cross- and triple-product properties and by noting that for groups, left and right inverses always coincide. Intuitively, because of the cross products, each row of is orthogonal to the non-corresponding two columns of (causing the off-diagonal terms of $\mathbf = \mathbf^\mathbf$ be zero). Dividing by : $\det\left(\mathbf\right) = \mathbf_0\cdot\left(\mathbf_1\times\mathbf_2\right)$ causes the diagonal elements of to be unity. For example, the first diagonal is: : $1 = \frac \mathbf\cdot\left(\mathbf_1\times\mathbf_2\right).$

### Inversion of 4 × 4 matrices

With increasing dimension, expressions for the inverse of get complicated. For , the Cayley–Hamilton method leads to an expression that is still tractable: :

## Blockwise inversion

Matrices can also be ''inverted blockwise'' by using the following analytic inversion formula: where , , and are matrix sub-blocks of arbitrary size. ( must be square, so that it can be inverted. Furthermore, and must be nonsingular.) This strategy is particularly advantageous if is diagonal and (the Schur complement of ) is a small matrix, since they are the only matrices requiring inversion. This technique was reinvented several times and is due to Hans Boltz (1923), who used it for the inversion of
geodetic Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equival ...
matrices, and Tadeusz Banachiewicz (1937), who generalized it and proved its correctness. The nullity theorem says that the nullity of equals the nullity of the sub-block in the lower right of the inverse matrix, and that the nullity of equals the nullity of the sub-block in the upper right of the inverse matrix. The inversion procedure that led to Equation () performed matrix block operations that operated on and first. Instead, if and are operated on first, and provided and are nonsingular, the result is Equating Equations () and () leads to where Equation () is the
Woodbury matrix identity In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury, says that the inverse of a rank-''k'' correction of some matrix can be computed by doing a rank-''k'' correction to the inverse of the origin ...
, which is equivalent to the binomial inverse theorem. If and are both invertible, then the above two block matrix inverses can be combined to provide the simple factorization By the Weinstein–Aronszajn identity, one of the two matrices in the block-diagonal matrix is invertible exactly when the other is. Since a blockwise inversion of an matrix requires inversion of two half-sized matrices and 6 multiplications between two half-sized matrices, it can be shown that a divide and conquer algorithm that uses blockwise inversion to invert a matrix runs with the same time complexity as the
matrix multiplication algorithm Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms efficient. Applications of matrix multiplication in computational problems are found in ...
that is used internally. Research into matrix multiplication complexity shows that there exist matrix multiplication algorithms with a complexity of operations, while the best proven lower bound is . This formula simplifies significantly when the upper right block matrix is the zero matrix. This formulation is useful when the matrices and have relatively simple inverse formulas (or pseudo inverses in the case where the blocks are not all square. In this special case, the block matrix inversion formula stated in full generality above becomes :$\begin \mathbf & \mathbf \\ \mathbf & \mathbf \end^ = \begin \mathbf^ & \mathbf \\ -\mathbf^\mathbf^ & \mathbf^ \end.$

## By Neumann series

If a matrix has the property that : $\lim_ \left(\mathbf I - \mathbf A\right)^n = 0$ then is nonsingular and its inverse may be expressed by a Neumann series: : $\mathbf A^ = \sum_^\infty \left(\mathbf I - \mathbf A\right)^n.$ Truncating the sum results in an "approximate" inverse which may be useful as a
preconditioner In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solving methods. Preconditioning is typically related to reducing ...
. Note that a truncated series can be accelerated exponentially by noting that the Neumann series is a geometric sum. As such, it satisfies : $\sum_^ \left(\mathbf I - \mathbf A\right)^n = \prod_^\left\left(\mathbf I + \left(\mathbf I - \mathbf A\right)^\right\right)$. Therefore, only matrix multiplications are needed to compute terms of the sum. More generally, if is "near" the invertible matrix in the sense that : $\lim_ \left\left(\mathbf I - \mathbf X^ \mathbf A\right\right)^n = 0 \mathrm \lim_ \left\left(\mathbf I - \mathbf A \mathbf X^\right\right)^n = 0$ then is nonsingular and its inverse is : $\mathbf A^ = \sum_^\infty \left\left(\mathbf X^ \left(\mathbf X - \mathbf A\right)\right\right)^n \mathbf X^~.$ If it is also the case that has rank 1 then this simplifies to : $\mathbf A^ = \mathbf X^ - \frac~.$

If is a matrix with
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
or rational coefficients and we seek a solution in arbitrary-precision rationals, then a -adic approximation method converges to an exact solution in , assuming standard matrix multiplication is used. The method relies on solving linear systems via Dixon's method of -adic approximation (each in ) and is available as such in software specialized in arbitrary-precision matrix operations, for example, in IML.

## Reciprocal basis vectors method

Given an square matrix , $1 \leq i,j \leq n$, with rows interpreted as vectors $\mathbf_ = x^ \mathbf_$ (
Einstein summation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
assumed) where the $\mathbf_$ are a standard
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
of
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
$\mathbb^$ ($\mathbf_ = \mathbf^, \mathbf_ \cdot \mathbf^ = \delta_i^j$), then using
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperco ...
(or
Geometric Algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the g ...
) we compute the reciprocal (sometimes called dual) column vectors: :$\mathbf^ = x_ \mathbf^ = \left(-1\right)^ \left(\mathbf_ \wedge\cdots\wedge \left(\right)_ \wedge\cdots\wedge\mathbf_\right) \cdot \left(\mathbf_ \wedge\ \mathbf_ \wedge\cdots\wedge\mathbf_\right)^$ as the columns of the inverse matrix Note that, the place "$\left(\right)_$" indicates that "$\mathbf_$" is removed from that place in the above expression for $\mathbf^$. We then have , where $\delta_^$ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
. We also have , as required. If the vectors $\mathbf_$ are not linearly independent, then $\left(\mathbf_ \wedge \mathbf_ \wedge\cdots\wedge\mathbf_\right) = 0$ and the matrix $\mathbf$ is not invertible (has no inverse).

# Derivative of the matrix inverse

Suppose that the invertible matrix A depends on a parameter ''t''. Then the derivative of the inverse of A with respect to ''t'' is given by : $\frac = - \mathbf^ \frac \mathbf^.$ To derive the above expression for the derivative of the inverse of A, one can differentiate the definition of the matrix inverse $\mathbf^\mathbf=\mathbf$ and then solve for the inverse of A: : $\frac = \frac\mathbf + \mathbf^\frac = \frac = \mathbf.$ Subtracting $\mathbf^\frac$ from both sides of the above and multiplying on the right by $\mathbf^$ gives the correct expression for the derivative of the inverse: : $\frac = - \mathbf^ \frac \mathbf^.$ Similarly, if $\varepsilon$ is a small number then : $\left\left(\mathbf + \varepsilon\mathbf\right\right)^ = \mathbf^ - \varepsilon \mathbf^ \mathbf \mathbf^ + \mathcal\left(\varepsilon^2\right)\,.$ More generally, if : $\frac = \sum_i g_i \left(\mathbf\right) \frach_i \left(\mathbf\right),$ then, : $f \left(\mathbf + \varepsilon\mathbf\right) = f \left(\mathbf\right) + \varepsilon\sum_i g_i \left(\mathbf\right) \mathbf h_i \left(\mathbf\right) + \mathcal\left\left(\varepsilon^2\right\right).$ Given a positive integer $n$, : $\begin \frac &= \sum_^n \mathbf^\frac\mathbf^,\\ \frac &= -\sum_^n \mathbf^\frac\mathbf^. \end$ Therefore, : $\begin \left(\mathbf + \varepsilon \mathbf\right)^ &= \mathbf^ + \varepsilon \sum_^n \mathbf^\mathbf\mathbf^ + \mathcal\left\left(\varepsilon^2\right\right),\\ \left(\mathbf + \varepsilon \mathbf\right)^ &= \mathbf^ - \varepsilon \sum_^n \mathbf^\mathbf\mathbf^ + \mathcal\left\left(\varepsilon^2\right\right). \end$

# Generalized inverse

Some of the properties of inverse matrices are shared by generalized inverses (for example, the
Moore–Penrose inverse In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Roge ...
), which can be defined for any ''m''-by-''n'' matrix.

# Applications

For most practical applications, it is ''not'' necessary to invert a matrix to solve a
system of linear equations In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in th ...
; however, for a unique solution, it ''is'' necessary that the matrix involved be invertible. Decomposition techniques like LU decomposition are much faster than inversion, and various fast algorithms for special classes of linear systems have also been developed.

## Regression/least squares

Although an explicit inverse is not necessary to estimate the vector of unknowns, it is the easiest way to estimate their accuracy, found in the diagonal of a matrix inverse (the posterior covariance matrix of the vector of unknowns). However, faster algorithms to compute only the diagonal entries of a matrix inverse are known in many cases.

## Matrix inverses in real-time simulations

Matrix inversion plays a significant role in
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great deal ...
, particularly in
3D graphics 3D computer graphics, or “3D graphics,” sometimes called CGI, 3D-CGI or three-dimensional computer graphics are graphics that use a three-dimensional representation of geometric data (often Cartesian) that is stored in the computer for th ...
rendering and 3D simulations. Examples include screen-to-world ray casting, world-to-subspace-to-world object transformations, and physical simulations.

## Matrix inverses in MIMO wireless communication

Matrix inversion also plays a significant role in the
MIMO In radio, multiple-input and multiple-output, or MIMO (), is a method for multiplying the capacity of a radio link using multiple transmission and receiving antennas to exploit multipath propagation. MIMO has become an essential element of wi ...
(Multiple-Input, Multiple-Output) technology in wireless communications. The MIMO system consists of ''N'' transmit and ''M'' receive antennas. Unique signals, occupying the same frequency band, are sent via ''N'' transmit antennas and are received via ''M'' receive antennas. The signal arriving at each receive antenna will be a linear combination of the ''N'' transmitted signals forming an ''N'' × ''M'' transmission matrix H. It is crucial for the matrix H to be invertible for the receiver to be able to figure out the transmitted information.