Invertible matrix

In linear algebra, an n-by-n (square) matrix A is called invertible or non-singular if there exists an n-by-n matrix B such that

$\mathbf{AB} = \mathbf{BA} = \mathbf{I}_n \$

where I_n denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. Linear algebra is the branch of Mathematics concerned with In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix If this is the case, then the matrix B is uniquely determined by A and is called the inverse of A, denoted by A^-1. It follows from the theory of matrices that if

$\mathbf{AB} = \mathbf{I} \$

for square matrices A and B, then also

$\mathbf{BA} = \mathbf{I} \ .$

Non-square matrices (m-by-n matrices for which m ≠ n) do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to If A is m-by-n and the rank of A is equal to n, then A has a left inverse: an n-by-m matrix B such that BA=I. The column rank of a matrix A is the maximal number of Linearly independent columns of A. If A has rank m, then it has a right inverse: an n-by-m matrix B such that AB=I.

While the most common case is that of matrices over the real or complex numbers, all these definitions can be given for matrices over any ring. In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0. ↔ In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

Over the field of real numbers, the set of singular n-by-n matrices, considered as a subset of $R^{n \times n}$ , is a null set, i. In Mathematics, a null set is a set that is negligible in some sense. e. , has Lebesgue measure zero. In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to In Mathematics, a null set is a set that is negligible in some sense. (This is true because singular matrices can be thought of as the roots of the polynomial function given by the determinant. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n ) This can be interpreted as saying that almost all n-by-n matrices are invertible. See also Generic property In Mathematics, the phrase almost all has a number of specialised uses Intuitively, this means that if you pick a random square matrix over the reals, the probability that it will be singular is zero. Probability is the likelihood or chance that something is the case or will happen 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. Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) In Numerical analysis, the condition number associated with a problem is a measure of that problem's amenability to digital computation that is hownumerically well-conditioned

Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.

1 Properties of invertible matrices
- 1.1 Proof for matrix product rule
2 Methods of matrix inversion
3 The derivative of the matrix inverse
4 The Moore-Penrose pseudoinverse
5 Matrix inverses in real-time simulations
6 See also
7 References
8 External links

Properties of invertible matrices

Let A be a square n by n matrix over a field $K$ (for example the field $R$ of real numbers). In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Then the following statements are equivalent:

A is invertible.
A is row-equivalent to the n-by-n identity matrix I_n. In Linear algebra, Gauss–Jordan elimination is a version of Gaussian elimination that puts zeros both above and below each Pivot element as it goes from In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main
A is column-equivalent to the n-by-n identity matrix I_n. In Linear algebra, Gauss–Jordan elimination is a version of Gaussian elimination that puts zeros both above and below each Pivot element as it goes from In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main
A has n pivot positions.
det A ≠ 0. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n
rank A = n. The column rank of a matrix A is the maximal number of Linearly independent columns of A.
The equation Ax = 0 has only the trivial solution x = 0 (i. e. , Null A = {0})
The equation Ax = b has exactly one solution for each b in $K n$ .
The columns of A are linearly independent. In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors
The columns of A span $K n$ (i. e. Col A = $K n$ ).
The columns of A form a basis of $K n$ . Basis vector redirects here For basis vector in the context of crystals see Crystal structure.
The linear transformation mapping x to Ax is a bijection from $K n$ to $K n$ . In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property
There is an n by n matrix B such that AB = I_n.
The transpose A^T is an invertible matrix. This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a
The matrix times its transpose, A^T × A is an invertible matrix.
The number 0 is not an eigenvalue of A. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit in that ring. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i

The inverse of an invertible matrix A is itself invertible, with

$\left(\mathbf{A}^{-1}\right)^{-1} = \mathbf{A}$ .

The inverse of an invertible matrix A multiplied by a non-zero scalar k yields the product of the inverse of both the matrix and the scalar

$\left(k\mathbf{A}\right)^{-1} = k^{-1}\mathbf{A}^{-1}$ .

For an invertible matrix A, the transpose of the inverse is the inverse of the transpose:

$(\mathbf{A}^\mathrm{T})^{-1} = (\mathbf{A}^{-1})^\mathrm{T} \,$

The product of two invertible matrices A and B of the same size is again invertible, with the inverse given by

$\left(\mathbf{AB}\right)^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}$

(note that the order of the factors is reversed. ) As a consequence, the set of invertible n-by-n matrices forms a group, known as the general linear group Gl(n). In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation

A matrix that is its own inverse, i. e. $\mathbf{A} = \mathbf{A}^{-1}$ and $\mathbf{A}^{2} = \mathbf{I}$ , is called an involution.

Proof for matrix product rule

If A₁, A₂, . . . , A_n are nonsingular square matrices over a field, then

$(\mathbf{A}_1\mathbf{A}_2\cdots \mathbf{A}_n)^{-1} = \mathbf{A}_n^{-1}\mathbf{A}_{n-1}^{-1}\cdots \mathbf{A}_1^{-1}$

It becomes evident why this is the case if one attempts to find an inverse for the product of the A_is from first principles, that is, that we wish to determine B such that

$(\mathbf{A}_1\mathbf{A}_2\cdots \mathbf{A}_n)\mathbf{B}=\mathbf{I}$

where B is the inverse matrix of the product. To remove A₁ from the product, we can then write

$\mathbf{A}_1^{-1}(\mathbf{A}_1\mathbf{A}_2\cdots \mathbf{A}_n)\mathbf{B}=\mathbf{A}_1^{-1}\mathbf{I}$

which would reduce the equation to

$(\mathbf{A}_2\mathbf{A}_3\cdots \mathbf{A}_n)\mathbf{B}=\mathbf{A}_1^{-1}\mathbf{I}$

Likewise, then, from

$\mathbf{A}_2^{-1}(\mathbf{A}_2\mathbf{A}_3\cdots \mathbf{A}_n)\mathbf{B}=\mathbf{A}_2^{-1}\mathbf{A}_1^{-1}\mathbf{I}$

which simplifies to

$(\mathbf{A}_3\mathbf{A}_4\cdots \mathbf{A}_n)\mathbf{B}=\mathbf{A}_2^{-1}\mathbf{A}_1^{-1}\mathbf{I}$

If one repeat the process up to A_n, the equation becomes

$\mathbf{B}=\mathbf{A}_n^{-1}\mathbf{A}_{n-1}^{-1}\cdots \mathbf{A}_2^{-1}\mathbf{A}_1^{-1}\mathbf{I}$

$\mathbf{B}=\mathbf{A}_n^{-1}\mathbf{A}_{n-1}^{-1}\cdots \mathbf{A}_2^{-1}\mathbf{A}_1^{-1}$

but B is the inverse matrix, i. e. $\mathbf{B} = (\mathbf{A}_1\mathbf{A}_2\cdots \mathbf{A}_n)^{-1}$ so the property is established.

Methods of matrix inversion

Gaussian elimination

Gauss–Jordan elimination is an algorithm that can be used to determine whether a given matrix is invertible and to find the inverse. In Linear algebra, Gauss–Jordan elimination is a version of Gaussian elimination that puts zeros both above and below each Pivot element as it goes from In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation An alternative is the LU decomposition which generates an upper and a lower triangular matrices which are easier to invert. In Linear algebra, the LU decomposition is a Matrix decomposition which writes a matrix as the product of a lower and upper Triangular matrix For special purposes, it may be convenient to invert matrices by treating mn-by-mn matrices as m-by-m matrices of n-by-n matrices, and applying one or another formula recursively (other sized matrices can be padded out with dummy rows and columns). For other purposes, a variant of Newton's method may be convenient (particularly when dealing with families of related matrices, so inverses of earlier matrices can be used to seed generating inverses of later matrices). In Numerical analysis, Newton's method (also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson) is perhaps the

Analytic solution

Writing another special 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. In Linear algebra, a minor of a matrix A is the Determinant of some smaller Square matrix, cut down from A by removing In Linear algebra, the adjugate or classical adjoint of a Square matrix is a matrix which plays a role similar to the inverse of a matrix; it To determine the inverse, we calculate a matrix of cofactors:

$\mathbf{A}^{-1}={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}}\left(\mathbf{C}_{ij}\right)^{\mathrm{T}}={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}}\left(\mathbf{C}_{ji}\right)={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}}\begin{pmatrix}\mathbf{C}_{11} & \mathbf{C}_{21} & \cdots & \mathbf{C}_{n1} \\\mathbf{C}_{12} & \mathbf{C}_{22} & \cdots & \mathbf{C}_{n2} \\\vdots & \vdots & \ddots & \vdots \\\mathbf{C}_{1n} & \mathbf{C}_{2n} & \cdots & \mathbf{C}_{nn} \\\end{pmatrix}$

where |A| is the determinant of A, C_ij is the matrix cofactor, and A^T represents the matrix transpose. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Linear algebra, a minor of a matrix A is the Determinant of some smaller Square matrix, cut down from A by removing This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a

For most practical applications, it is not necessary to invert a matrix to solve a system of linear equations; however, for a unique solution, it is necessary that the matrix involved be invertible. In Mathematics, a system of linear equations (or linear system) is a collection of Linear equations involving the same set of Variables For example

Decomposition techniques like LU decomposition are much faster than inversion, and various fast algorithms for special classes of linear systems have also been developed. In Linear algebra, the LU decomposition is a Matrix decomposition which writes a matrix as the product of a lower and upper Triangular matrix

Inversion of 2×2 matrices

The cofactor equation listed above yields the following result for 2×2 matrices. Inversion of these matrices can be done easily as follows: ^[1]

$\mathbf{A}^{-1} = \begin{bmatrix}a & b \\ c & d \\\end{bmatrix}^{-1} =\frac{1}{ad - bc} \begin{bmatrix}\,\,\,d & \!\!-b \\ -c & \,a \\\end{bmatrix}.$

This is possible because 1/(ad-bc) is the reciprocal of the determinant of the matrix in question, and the same strategy could be used for other matrix sizes.

Blockwise inversion

Matrices can also be inverted blockwise by using the following analytic inversion formula:

$\begin{bmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \end{bmatrix}^{-1} = \begin{bmatrix} \mathbf{A}^{-1}+\mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1} & -\mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1} \\ -(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1} & (\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1} \end{bmatrix}$

$(1)\,$

where A, B, C and D are matrix sub-blocks of arbitrary size. In the mathematical discipline of Matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices (A and D must, of course, be square, so that they can be inverted). This strategy is particularly advantageous if A is diagonal and D−CA^-1B (the Schur complement of A) is a small matrix, since they are the only matrices requiring inversion. In Linear algebra and the theory of matrices,the Schur complement of a block of a matrix within alarger matrix is defined as follows This technique was reinvented several times and is due to Hans Bolz (1923), who used it for the inversion of geodetic matrices, and Tadeusz Banachiewicz (1937), who generalized it and proved its correctness.

The inversion procedure that led to Equation (1) performed matrix block operations that operated on C and D first. Instead, if A and B are operated on first, the result is

$\begin{bmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \end{bmatrix}^{-1} = \begin{bmatrix} (\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1} & -(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}\mathbf{BD}^{-1} \\ -\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1} & \mathbf{D}^{-1}+\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}\mathbf{BD}^{-1}\end{bmatrix}$

$(2)\,$

Equating Equations (1) and (2) leads to

$(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1} = \mathbf{A}^{-1}+\mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1}\,$

$(3)\,$

$(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}\mathbf{BD}^{-1} = \mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\,$

$\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1} = (\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1}\,$

$\mathbf{D}^{-1}+\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}\mathbf{BD}^{-1} = (\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\,$

where Equation (3) is the matrix inversion lemma, which is equivalent to the binomial inverse theorem. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- In Mathematics, the binomial inverse theorem is useful for expressing matrix inverses in different ways

Proof of matrix inversion lemma

First, multiply the RHS of Equation (3) by the inverse of the LHS to get

$\mathbf{I}=\mathbf{I}-\mathbf{BD}^{-1}\mathbf{CA}^{-1}+\left(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C}\right)\left(\mathbf{A}^{-1}\mathbf{B}\right)\left(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B}\right)^{-1}\mathbf{CA}^{-1}$

Note that

$\begin{align}&\left(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C}\right)\left(\mathbf{A}^{-1}\mathbf{B}\right)\left(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B}\right)^{-1}\mathbf{C}\mathbf{A}^{-1} \\=&\left(\mathbf{B}-\mathbf{BD}^{-1}\mathbf{CA}^{-1}\mathbf{B}\right)\left(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B}\right)^{-1}\mathbf{C}\mathbf{A}^{-1} \\=&\mathbf{B}\underbrace{\left(\mathbf{I}-\mathbf{D}^{-1}\mathbf{CA}^{-1}\mathbf{B}\right)\left(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B}\right)^{-1}}_\mathbf{P}\mathbf{C}\mathbf{A}^{-1}\end{align}$

If we can show that $\mathbf{P}=\mathbf{D}^{-1}$ , then the $\mathbf{B}\mathbf{D}^{-1}\mathbf{C}\mathbf{A}^{-1}$ terms would be cancelled out. This is accomplished by noting

$\left(\mathbf{I}-\mathbf{D}^{-1}\mathbf{CA}^{-1}\mathbf{B}\right)\left(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B}\right)^{-1}=\mathbf{D}^{-1}\left(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B}\right)\left(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B}\right)^{-1}=\mathbf{D}^{-1}$

Hence we have shown that $\mathbf{P}$ is indeed equal to $\mathbf{D}^{-1}$ . After the cancellation of the $\mathbf{B}\mathbf{D}^{-1}\mathbf{C}\mathbf{A}^{-1}$ term, all there is left is the identity matrix; and the proof is completed.

The derivative of the matrix inverse

Suppose that the matrix A depends on a parameter t. Then the derivative of the inverse of A with respect to t is given by

$\frac{\mathrm{d}\mathbf{A}^{-1}}{\mathrm{d}t} = - \mathbf{A}^{-1} \frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t} \mathbf{A}^{-1}.$

This formula can be found by differentiating the identity

$\mathbf{A}^{-1}\mathbf{A} = \mathbf{I}.\,$

The Moore-Penrose pseudoinverse

Some of the properties of inverse matrices are shared by (Moore-Penrose) pseudoinverses, which can be defined for any m-by-n matrix. In Mathematics, and in particular Linear algebra, the pseudoinverse A^+ of an m \times n matrix A is a generalization

Matrix inverses in real-time simulations

Matrix inversion plays a significant role in computer graphics, particularly in 3D graphics rendering and 3D simulations. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data 3D computer graphics (in contrast to 2D computer graphics) are graphics that use a three-dimensional representation of geometric data that is stored in the computer Examples include screen-to-world ray casting, world-to-subspace-to-world object transformations, and physical simulations. The problem is usually the numerical complexity of calculating the inverses of 3×3 and 4×4 matrices. Compared to matrix multiplication or creation of rotation matrices, matrix inversion is several orders of magnitude slower. There are existing solutions which use hand-crafted assembly language routines and SIMD processor extensions (SSE, SSE2, Altivec) that address this problem and which achieve a performance improvement of as much as five times. See the terminology section below for information regarding inconsistent use of the terms assembly and assembler In Computing, SIMD ( S ingle I nstruction M ultiple D ata is a technique employed to achieve data level parallelism as in a Vector

References

Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. In the mathematical discipline of Linear algebra, a matrix decomposition is a Factorization of a matrix into some Canonical form. In Mathematics, the binomial inverse theorem is useful for expressing matrix inverses in different ways In Linear algebra, the singular value decomposition ( SVD) is an important factorization of a rectangular real or complex matrix In Linear algebra, the LU decomposition is a Matrix decomposition which writes a matrix as the product of a lower and upper Triangular matrix Thomas H Cormen is the co-author of Introduction to Algorithms, along with Charles Leiserson, Ron Rivest, and Cliff Stein. Charles Eric Leiserson is a Computer scientist, specializing in the theory of Parallel computing and Distributed computing, and particularly practical Ronald Linn Rivest (born 1947, Schenectady, New York) is a cryptographer. Clifford Stein, a Computer scientist, is currently a professor of Industrial engineering and Operations research at Columbia University Introduction to Algorithms, Second Edition. Introduction to Algorithms is a book by Thomas H Cormen, Charles E MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 28. 4: Inverting matrices, pp. 755–760.

^ Strang, Gilbert (2006). William Gilbert Strang, usually known as simply Gilbert Strang, is a renowned American Mathematician, with contributions to finite element theory Linear Algebra and Its Applications. Thomson Brooks/Cole, p. 46. ISBN 0-03-010567-6.

External links

MIT Linear Algebra Lecture on Inverse Matrices at Google Video, from MIT OpenCourseWare
LAPACK is a collection of FORTRAN subroutines for solving dense linear algebra problems
ALGLIB includes a partial port of the LAPACK to C++, C#, Delphi, etc.
Online Inverse Matrix Calculator using AJAX
Inverse Matrix Calculator
Moore Penrose Pseudoinverse
Inverse of a Matrix Notes
Module for the Matrix Inverse
Calculator for Singular or Non-Square Matrix Inverse
Derivative of inverse matrix on PlanetMath

PlanetMath is a free, collaborative online Mathematics Encyclopedia.

Dictionary

-noun

(mathematics) a square matrix which, when multiplied by another (in either order), yields the identity matrix

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