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In linear algebra an nbyn (square) matrix A is called invertible (some authors use nonsingular or nondegenerate) if there exists an nbyn matrix B such that
where I_{n} denotes the nbyn identity matrix and the multiplication used is ordinary matrix multiplication. 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
for finite square matrices A and B, then also
Nonsquare matrices (mbyn 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. If A is mbyn and the rank of A is equal to n, then A has a left inverse: an nbym matrix B such that BA = I. If A has rank m, then it has a right inverse: an nbym matrix B such that AB = I.
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. Singular matrices are rare in the sense that if you pick a random square matrix over a continuous uniform distribution on its entries, it will almost surely not be singular.
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 commutative ring. However, in this case the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a much stricter requirement than being nonzero. The conditions for existence of leftinverse resp. rightinverse are more complicated since a notion of rank does not exist over rings.
Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.
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Let A be a square n by n matrix over a field K (for example the field R of real numbers). The following statements are equivalent:
Furthermore, the following properties hold for an invertible matrix A:
A matrix that is its own inverse, i.e. A = A^{1} and A^{2} = I, is called an involution.
Over the field of real numbers, the set of singular nbyn matrices, considered as a subset of R^{n×n}, is a null set, i.e., has Lebesgue measure zero. This is true because singular matrices are the roots of the polynomial function in the entries of the matrix given by the determinant. Thus in the language of measure theory, almost all nbyn matrices are invertible.
Furthermore the nbyn invertible matrices are a dense open set in the topological space of all nbyn matrices. Equivalently, the set of singular matrices is closed and nowhere dense in the space of nbyn matrices.
In practice however, one may encounter noninvertible matrices. And in numerical calculations, matrices which are invertible, but close to a noninvertible matrix, can still be problematic; such matrices are said to be illconditioned.
Gauss–Jordan elimination is an algorithm that can be used to determine whether a given matrix is invertible and to find the inverse. An alternative is the LU decomposition which generates upper and lower triangular matrices which are easier to invert. For special purposes, it may be convenient to invert matrices by treating m·nbym·n matrices as mbym matrices of nbyn 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. Newton's method is particularly useful when dealing with families of related matrices: 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. 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.
If matrix A can be eigendecomposed and if none of its eigenvalues are zero, then A is nonsingular and its inverse is given by
Furthermore, because Λ is a diagonal matrix, its inverse is easy to calculate:
If matrix A is positive definite, then its inverse can be obtained as
where L is the lower triangular Cholesky decomposition of A.
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:
where A is the determinant of A, C_{ij} is the matrix of cofactors, and C^{T} represents the matrix transpose.
The cofactor equation listed above yields the following result for 2×2 matrices. Inversion of these matrices can be done easily as follows:^{[2]}
This is possible because 1/(adbc) is the reciprocal of the determinant of the matrix in question, and the same strategy could be used for other matrix sizes.
A computationally efficient 3x3 matrix inversion is given by
where the determinant of A can be computed by applying the rule of Sarrus as follows:
If the determinant is nonzero, the matrix is invertible, with the elements of the above matrix on the right side given by
The general 3×3 inverse can be expressed concisely in terms of the cross product and triple product:
If a matrix (consisting of three column vectors, , , and ) is invertible, its inverse is given by
Note that is equal to the triple product of , , and —the volume of the parallelepiped formed by the rows or columns:
The correctness of the formula can be checked by using cross and tripleproduct 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 noncorresponding two columns of (causing the offdiagonal terms of be zero). Dividing by
causes the diagonal elements of to be unity. For example, the first diagonal is:
Matrices can also be inverted blockwise by using the following analytic inversion formula:

where A, B, C and D are matrix subblocks of arbitrary size. (A and D must, of course, be square, so that they can be inverted. Furthermore, this is true if and only if A and D−CA^{−1}B are nonsingular ^{[3]} ). This strategy is particularly advantageous if A is diagonal and D−CA^{−1}B (the Schur complement of A) 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),^{[citation needed]} who used it for the inversion of geodetic matrices, and Tadeusz Banachiewicz (1937), who generalized it and proved its correctness.
The nullity theorem says that the nullity of A equals the nullity of the subblock in the lower right of the inverse matrix, and that the nullity of B equals the nullity of the subblock in the upper right of the inverse matrix.
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, and provided D and A−BD^{−1}C are nonsingular ,^{[4]} the result is

Equating Equations (1) and (2) leads to

where Equation (3) is the matrix inversion lemma, which is equivalent to the binomial inverse theorem.
Since a blockwise inversion of an n×n matrix requires inversion of two halfsized matrices and 6 mulitplications between two halfsized matrices, and since matrix multiplication algorithm has a lower bound of Ω(n^{2} log n) operations^{[5]}, 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 that is used internally.
If a matrix A has the property that
then A is nonsingular and its inverse may be expressed by a Neumann series:^{[6]}
Truncating the sum results in an "approximate" inverse which may be useful as a preconditioner.
More generally, if A is "near" the invertible matrix X in the sense that
then A is nonsingular and its inverse is
If it is also the case that AX has rank 1 then this simplifies to
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
To derive the above expression for the derivative of the inverse of A, one can differentiate the definition of the matrix inverse and then solve for the inverse of A:
Subtracting from both sides of the above and multiplying on the right by gives the correct expression for the derivative of the inverse:
Similarly, if is a small number then
Some of the properties of inverse matrices are shared by Moore–Penrose pseudoinverses, which can be defined for any mbyn matrix.
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.
Decomposition techniques like LU decomposition are much faster than inversion, and various fast algorithms for special classes of linear systems have also been developed.
Matrix inversion plays a significant role in computer graphics, particularly in 3D graphics rendering and 3D simulations. Examples include screentoworld ray casting, worldtosubspacetoworld object transformations, and physical simulations.
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