Invertible matrix
In linear algebra, an invertible matrix is a square matrix that has an inverse. In other words, if a matrix is invertible, it can be multiplied by its inverse matrix to yield the identity matrix. Invertible matrices are the same size as their inverse.
The inverse of a matrix represents the inverse operation, meaning if a matrix is applied to a particular vector, followed by applying the matrix's inverse, the result is the original vector.
Definition
An -by- square matrix is called invertible if there exists an -by- square matrix such thatwhere denotes the -by- 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. Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix.Examples
Consider the following 2-by-2 matrix:The matrix is invertible, as it has inverse which can be confirmed by computing
To check that it is invertible without finding an inverse, can be computed, which is non-zero.
On the other hand, this is a non-invertible matrix:
We can see the rank of this 2-by-2 matrix is 1, which is, so it is non-invertible. Additionally, we can compute that 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
Gaussian elimination 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. 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:
The first step to compute its inverse is to create the augmented matrix
Call the first row of this matrix and the second row. Then, add row 1 to row 2 This yields
Next, subtract row 2, multiplied by 3, from row 1 which yields
Finally, multiply row 1 by −1 and row 2 by 2 This yields the identity matrix on the left side and the inverse matrix on the right:
Thus,
It works because the process of Gaussian elimination can be viewed as a sequence of applying left matrix multiplication using elementary row operations using elementary matrices, such as
Applying right-multiplication using we get And the right side which is the inverse we want.
To obtain we create the augmented matrix by combining with and applying Gaussian elimination. 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 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.
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. That 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 from imperfect computer arithmetic.
Cayley–Hamilton method
The Cayley–Hamilton theorem allows the inverse of to be expressed in terms of, traces and powers of :where is size 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
That is described in more detail under Cayley–Hamilton method.
Eigendecomposition
If matrix can be eigendecomposed, and if none of its eigenvalues are zero, then is invertible and its inverse is given bywhere is the square matrix whose th column is the eigenvector of, and is the diagonal matrix whose diagonal entries 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 is positive definite, then its inverse can be obtained aswhere is the lower triangular Cholesky decomposition of, and denotes the conjugate transpose of.
Analytic solution
Writing the transpose of the matrix of cofactors, known as an adjugate matrix, may also be an efficient way to calculate the inverse of small matrices, but the recursive method is inefficient for large matrices. To determine the inverse, we calculate a matrix of cofactors:so that
where is the determinant of, is the matrix of cofactors, and 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 entries of the intermediary matrix on the right side above given by
The determinant of 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,, 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 entries of to be unity. For example, the first diagonal is:
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
Letwhere,, and are matrix sub-blocks of arbitrary size and is the Schur complement of.
Matrices can also be inverted blockwise by using the analytic inversion formula:
The strategy is particularly advantageous if is diagonal and is a small matrix, since they are the only matrices requiring inversion.
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 the upper-left sub-matrices of Equations and leads to
where Equation is the Woodbury matrix identity, 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.
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, if and only if is invertible. By writing matrix inversion can be reduced to inverting symmetric matrices and 2 additional matrix multiplications, because the positive definite matrix satisfies the invertibility condition for its left upper block.
Those formulas together allow to construct a divide and conquer algorithm that uses blockwise inversion of associated symmetric matrices to invert a matrix with the same time complexity as the matrix multiplication algorithm 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.
By Neumann series
If a matrix has the property thatthen 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 is "near" the invertible matrix in the sense that
then is nonsingular and its inverse is
If it is also the case that has rank 1 then this simplifies to
''p''-adic approximation
If is a matrix with integer or rational entries, and we seek a solution in arbitrary-precision rationals, 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 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.
Properties
Singularity
Over a field, a square matrix that is not invertible is called singular or degenerate. A square matrix with entries in a field is singular if and only if its determinant is zero.Invertible matrix theorem
Let be a square -by- matrix over a field . The following statements are equivalent, i.e., they are either all true or all false for any given matrix:- is invertible, i.e. it has an inverse under matrix multiplication, i.e., there exists a such that.
- The linear transformation mapping to is invertible, i.e., it has an inverse under function composition.
- The transpose is an invertible matrix.
- is row-equivalent to the -by- identity matrix.
- is column-equivalent to the -by- identity matrix.
- has pivot positions.
- has full rank:.
- has a trivial kernel:
- The linear transformation mapping to is bijective; that is, the equation has exactly one solution for each in.
- The columns of form a basis of.
- The rows of form a basis of.
- The determinant of is nonzero:. In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit of that ring.
- The number 0 is not an eigenvalue of.
- The matrix can be expressed as a finite product of elementary matrices.
Other properties
Furthermore, the following properties hold for an invertible matrix :- for nonzero scalar
- if has orthonormal columns, where denotes the Moore–Penrose inverse and is a vector
- For any invertible -by- matrices and, More generally, if are invertible -by- matrices, then
- Left and right inverses are equal. That is, if and then.
A matrix that is its own inverse 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 and, then also
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. That is true because singular matrices are the roots of the determinant function. It is a continuous function because it is a polynomial in the entries of the matrix. Thus in the language of measure theory, almost all -by- matrices are invertible.Furthermore, the set of -by- invertible matrices is open and dense in the topological space of all -by- matrices. Equivalently, the set of singular matrices is closed and nowhere dense in the space of -by- matrices.
In practice, however, non-invertible matrices may be encountered. In numerical calculations, matrices that are invertible but close to a non-invertible matrix may still be problematic and are said to be ill-conditioned.
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 using the product rule, and then solve for the derivative of the inverse of A:
Subtracting from both ends of this formula, and multiplying on the right by finishes the derivation.
If is a small number then the derivative formula gives:
Given a positive integer,
In particular,
Generalizations
Non-square matrices
Non-square matrices, i.e. -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.Some of the properties of inverse matrices are shared by generalized inverses, which can be defined for any m-by-n matrix.
In Abstract algebra
While the most common case is that of matrices over the real or complex numbers, all of those definitions can be given for matrices over any algebraic structure equipped with addition and multiplication. However, in the case of a 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 it 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 of degree, denoted.
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 for the matrix involved to 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.