Invertible matrix
In linear algebra, an n-by-n square matrix is called invertible if there exists an n-by-n square matrix such that
where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by and is called the inverse of, denoted by .
A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if 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 do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If is m-by-n 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.
Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix.
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. 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 form a group, the general linear group of degree n.
Properties
The invertible matrix theorem
Let A be a square n by n matrix over a field K. The following statements are equivalent, that is, for any given matrix they are either all true or all false:Other properties
Furthermore, the following properties hold for an invertible matrix A:- ^{−1} = A;
- ^{−1} = k^{−1}A^{−1} for nonzero scalar k;
- ^{+} = x^{+}A^{−1} if A has orthonormal columns, where ^{+} denotes the Moore–Penrose inverse and x is a vector;
- ^{−1} = ^{T};
- For any invertible n-by-n matrices A and B, ^{−1} = B^{−1}A^{−1}. More generally, if A_{1},..., A_{k} are invertible n-by-n matrices, then ^{−1} = A'A⋯A'A;
- det A^{−1} = ^{−1}.
A matrix that is its own inverse, that is, 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
In relation to the identity matrix
It follows from the associativity of matrix multiplication that iffor finite square matrices A and B, then also
Density
Over the field of real numbers, the set of singular n-by-n matrices, considered as a subset of R^{n×n}, is a null set, that is, has Lebesgue measure zero. 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, almost all n-by-n matrices are invertible.Furthermore, the n-by-n invertible matrices are a dense open set in the topological space of all n-by-n matrices. Equivalently, the set of singular matrices is closed and nowhere dense in the space of n-by-n 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.
Examples
Consider the following 2-by-2 matrix:The matrix is invertible. To check this, one can compute that, which is non-zero.
As an example of a non-invertible, or singular, matrix, consider the matrix
The determinant of is 0, which is a necessary and sufficient condition for a matrix to be non-invertible.
Methods of matrix inversion
Gaussian 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.Newton's method
A generalization of Newton's method as used for a multiplicative inverse algorithm may be convenient, if it is convenient to find a suitable starting seed: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 det, traces and powers of :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 satisfying the linear Diophantine equation
The formula can be rewritten in terms of complete Bell polynomials of arguments as
Eigendecomposition
If matrix A can be eigendecomposed, and if none of its eigenvalues are zero, then A is invertible and its inverse is given bywhere is the square matrix whose i-th column is the eigenvector of, and is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, that is,. If
is symmetric, is guaranteed to be an orthogonal matrix, therefore. Furthermore, because is a diagonal matrix, its inverse is easy to calculate:
Cholesky decomposition
If matrix A is positive definite, then its inverse can be obtained aswhere L is the lower triangular Cholesky decomposition of A, and L* denotes the conjugate transpose of L.
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:so that
where |A| is the determinant of A, C is the matrix of cofactors, and C^{T} represents the matrix transpose.
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: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.
If the determinant is non-zero, the matrix is invertible, with the elements of the intermediary matrix on the right side above given by
The determinant of A can be computed by applying the rule of Sarrus as follows:
The Cayley–Hamilton decomposition gives
The general inverse can be expressed concisely in terms of the cross product and triple product. If a matrix is invertible, its inverse is given by
The determinant of A,, 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 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 . Dividing by
causes the diagonal elements of to be unity. For example, the first diagonal is:
Inversion of 4 × 4 matrices
With increasing dimension, expressions for the inverse of A 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 A, B, C and D are matrix sub-blocks of arbitrary size. This strategy is particularly advantageous if A is diagonal and is a small matrix, since they are the only matrices requiring inversion.
This technique was reinvented several times and is due to Hans Boltz, who used it for the inversion of geodetic matrices, and Tadeusz Banachiewicz, who generalized it and proved its correctness.
The nullity theorem says that the nullity of A equals the nullity of the sub-block in the lower right of the inverse matrix, and that the nullity of B 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 C and D first. Instead, if A and B are operated on first, and provided D and are nonsingular, the result is
Equating Equations and leads to
where Equation is the Woodbury matrix identity, which is equivalent to the binomial inverse theorem.
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 that is used internally. 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
By Neumann series
If a matrix A has the property thatthen A is nonsingular and its inverse may be expressed by a Neumann series:
Truncating the sum results in an "approximate" inverse which may be useful as a preconditioner. Note that a truncated series can be accelerated exponentially by noting that the Neumann series is a geometric sum. As such, it satisfies
Therefore, only matrix multiplications are needed to compute terms of the sum.
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 has rank 1 then this simplifies to
''p''-adic approximation
If A is a matrix with integer or rational coefficients and we seek a solution in arbitrary-precision rationals, then a p-adic approximation method converges to an exact solution in, assuming standard matrix multiplication is used. The method relies on solving n linear systems via Dixon's method of p-adic approximation 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,, with rows interpreted as vectors where the are a standard orthonormal basis of Euclidean space , then using Clifford algebra we compute the reciprocal column vectors as the columns of the inverse matrix. Note that, the place "" indicates that "" is removed from that place in the above expression for. We then have, where is the Kronecker delta. We also have, as required. If the vectors are not linearly independent, then and the matrix is not invertible.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 byTo 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
More generally, if
then,
Given a positive integer,
Therefore,
Generalized inverse
Some of the properties of inverse matrices are shared by generalized inverses, 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; 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.