Invertible matrix
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In linear algebra, an nbyn square matrix A is called invertible (also nonsingular, nondegenerate or —rarely used— regular), if there exists an nbyn square matrix B such that
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. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any bounded 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. Nonsquare matrices, i.e. 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, (n ≤ m), then A has a left inverse, an nbym matrix B such that BA = I_{n}. If A has rank m (m ≤ n), then it has a right inverse, an nbym matrix B such that AB = I_{m}.
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 algebraic structure equipped with addition and multiplication (i.e. rings). 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 leftinverse or rightinverse are more complicated, since a notion of rank does not exist over rings.
The set of n × n invertible matrices together with the operation of matrix multiplication and entries from ring R form a group, the general linear group of degree n, denoted GL_{n}(R).
Properties[edit]
The invertible matrix theorem[edit]
Let A be a square nbyn matrix over a field K (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:^{[2]}
 The matrix A has a left inverse under matrix multiplication (that is, there exists a B such that BA = I)
 The matrix A has a right inverse under matrix multiplication (that is, there exists a C such that AC = I).
 A is invertible, i.e. it has an inverse under matrix multiplication (that is, there exists a B such that AB = I_{n} = BA).
 The linear transformation mapping x to Ax has a left inverse under function composition.
 The linear transformation mapping x to Ax has a right inverse under function composition.
 The linear transformation mapping x to Ax is invertible, that is, has an inverse under function composition.
 A is rowequivalent to the nbyn identity matrix I_{n}.
 A is columnequivalent to the nbyn identity matrix I_{n}.
 A has n pivot positions.
 A has full rank: rank A = n.
 A has a trivial kernel: ker(A) = {0}.
 The linear transformation mapping x to Ax is surjective; that is, the equation Ax = b has at least one solution for each b in K^{n}.
 The linear transformation mapping x to Ax is injective; that is, the equation Ax = b has at most one solution for each b in K^{n}.
 The linear transformation mapping x to Ax is bijective; that is, the equation Ax = b has exactly one solution for each b in K^{n}.
 The columns of A are linearly independent.
 The rows of A are linearly independent.
 The columns of A span K^{n}.
 The rows of A span K^{n}.
 The columns of A form a basis of K^{n}.
 The rows of A form a basis of K^{n}.
 The determinant of A is nonzero: det A ≠ 0. (In general, a square matrix over a commutative ring is invertible if and only if its determinant is an element of that ring with a multiplicative inverse (i.e. a unit)
 The number 0 is not an eigenvalue of A. (More generally, a number is an eigenvalue of A if the matrix is singular, where I is the identity matrix.)
 The transpose A^{T} is an invertible matrix.
 The matrix A can be expressed as a finite product of elementary matrices.
Other properties[edit]
Furthermore, the following properties hold for an invertible matrix A:
 for nonzero scalar k
 if A has orthonormal columns, where ^{+} denotes the Moore–Penrose inverse and x is a vector
 For any invertible nbyn matrices A and B, More generally, if are invertible nbyn matrices, then
The rows of the inverse matrix V of a matrix U are orthonormal to the columns of U (and vice versa interchanging rows for columns). To see this, suppose that UV = VU = I where the rows of V are denoted as and the columns of U as for Then clearly, the Euclidean inner product of any two This property can also be useful in constructing the inverse of a square matrix in some instances, where a set of orthogonal vectors (but not necessarily orthonormal vectors) to the columns of U are known. In which case, one can apply the iterative Gram–Schmidt process to this initial set to determine the rows of the inverse V.
A matrix that is its own inverse (i.e., a matrix A such that A = A^{−1}, and consequently A^{2} = I), is called an involutory matrix.
In relation to its adjugate[edit]
The adjugate of a matrix A can be used to find the inverse of A as follows:
If A is an invertible matrix, then
In relation to the identity matrix[edit]
It follows from the associativity of matrix multiplication that if
for finite square matrices A and B, then also
 ^{[3]}
Density[edit]
Over the field of real numbers, the set of singular nbyn matrices, considered as a subset of 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 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.
Examples[edit]
An example with rank of n − 1 is a noninvertible matrix
We can see the rank of this 2by2 matrix is 1, which is n − 1 ≠ n, so it is noninvertible.
Consider the following 2by2 matrix:
The matrix is invertible. To check this, one can compute that , which is nonzero.
As an example of a noninvertible, or singular, matrix, consider the matrix
The determinant of is 0, which is a necessary and sufficient condition for a matrix to be noninvertible.
Methods of matrix inversion[edit]
Gaussian elimination[edit]
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,
The reason it works is that 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 rightmultiplication using we get And the right side which is the inverse we want.
To obtain we create the augumented matrix by combining A with I and applying Gaussian elimination. The two portions will be transformed using the same sequence of elementary row operations. When the left portion becomes I, the right portion applied the same elementary row operation sequence will become A^{−1}.
Newton's method[edit]
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.^{[4]}^{[5]} Byte magazine summarised one of their approaches.^{[6]}
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[edit]
The Cayley–Hamilton theorem allows the inverse of A to be expressed in terms of det(A), traces and powers of A:^{[7]}
where n is dimension of A, and tr(A) is the trace of matrix A given by the sum of the main diagonal. The sum is taken over s 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[edit]
If matrix A can be eigendecomposed, and if none of its eigenvalues are zero, then A is invertible and its inverse is given by
where Q is the square (N × N) matrix whose ith column is the eigenvector of A, and Λ is the diagonal matrix whose diagonal entries are the corresponding eigenvalues, that is, If A is symmetric, Q is guaranteed to be an orthogonal matrix, therefore Furthermore, because Λ is a diagonal matrix, its inverse is easy to calculate:
Cholesky decomposition[edit]
If matrix A is positive definite, then its inverse can be obtained as
where L is the lower triangular Cholesky decomposition of A, and L* denotes the conjugate transpose of L.
Analytic solution[edit]
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[edit]
The cofactor equation listed above yields the following result for 2 × 2 matrices. Inversion of these matrices can be done as follows:^{[8]}
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.
The Cayley–Hamilton method gives
Inversion of 3 × 3 matrices[edit]
A computationally efficient 3 × 3 matrix inversion is given by
(where the scalar A is not to be confused with the matrix A).
If the determinant is nonzero, the matrix is invertible, with the entries 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 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
The determinant of A, det(A), is equal to the triple product of x_{0}, x_{1}, and x_{2}—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 A^{–1} is orthogonal to the noncorresponding two columns of A (causing the offdiagonal terms of be zero). Dividing by
causes the diagonal entries of I = A^{−1}A to be unity. For example, the first diagonal is:
Inversion of 4 × 4 matrices[edit]
With increasing dimension, expressions for the inverse of A get complicated. For n = 4, the Cayley–Hamilton method leads to an expression that is still tractable:
Blockwise inversion[edit]
Matrices can also be inverted blockwise by using the following analytic inversion formula:^{[9]}

(1)
where A, B, C and D are matrix subblocks of arbitrary size. (A must be square, so that it can be inverted. Furthermore, A and D − CA^{−1}B must be nonsingular.^{[10]}) 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,^{[11]} the result is

(2)
Equating Equations (1) and (2) leads to

(3)
where Equation (3) is the Woodbury matrix identity, which is equivalent to the binomial inverse theorem.
If A and D are both invertible, then the above two block matrix inverses can be combined to provide the simple factorization

(2)
By the Weinstein–Aronszajn identity, one of the two matrices in the blockdiagonal matrix is invertible exactly when the other is.
Since a blockwise inversion of an n × n matrix requires inversion of two halfsized matrices and 6 multiplications between two halfsized 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.^{[12]} Research into matrix multiplication complexity shows that there exist matrix multiplication algorithms with a complexity of O(n^{2.3727}) operations, while the best proven lower bound is Ω(n^{2} log n).^{[13]}
This formula simplifies significantly when the upper right block matrix B is the zero matrix. This formulation is useful when the matrices A and D 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[edit]
If a matrix A has the property that
then A is nonsingular and its inverse may be expressed by a Neumann series:^{[14]}
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 2L − 2 matrix multiplications are needed to compute 2^{L} 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 A − X has rank 1 then this simplifies to
padic approximation[edit]
If A is a matrix with integer or rational entries and we seek a solution in arbitraryprecision rationals, then a padic approximation method converges to an exact solution in O(n^{4} log^{2} n), assuming standard O(n^{3}) matrix multiplication is used.^{[15]} The method relies on solving n linear systems via Dixon's method of padic approximation (each in O(n^{3} log^{2} n)) and is available as such in software specialized in arbitraryprecision matrix operations, for example, in IML.^{[16]}
Reciprocal basis vectors method[edit]
Given an n × n square matrix , , with n rows interpreted as n vectors (Einstein summation assumed) where the are a standard orthonormal basis of Euclidean space (), then using Clifford algebra (or geometric algebra) we compute the reciprocal (sometimes called dual) 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 (has no inverse).
Derivative of the matrix inverse[edit]
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^{[17]}
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
More generally, if
then,
Given a positive integer ,
Therefore,
Generalized inverse[edit]
Some of the properties of inverse matrices are shared by generalized inverses (for example, the Moore–Penrose inverse), which can be defined for any mbyn matrix.^{[18]}
Applications[edit]
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.
Regression/least squares[edit]
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.^{[19]}
Matrix inverses in realtime simulations[edit]
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.
Matrix inverses in MIMO wireless communication[edit]
Matrix inversion also plays a significant role in the MIMO (MultipleInput, MultipleOutput) 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.
See also[edit]
References[edit]
 ^ Axler, Sheldon (18 December 2014). Linear Algebra Done Right. Undergraduate Texts in Mathematics (3rd ed.). Springer Publishing (published 2015). p. 296. ISBN 9783319110790.
 ^ Weisstein, Eric W. "Invertible Matrix Theorem". mathworld.wolfram.com. Retrieved 20200908.
 ^ Horn, Roger A.; Johnson, Charles R. (1985). Matrix Analysis. Cambridge University Press. p. 14. ISBN 9780521386326..
 ^ Pan, Victor; Reif, John (1985), Efficient Parallel Solution of Linear Systems, Proceedings of the 17th Annual ACM Symposium on Theory of Computing, Providence: ACM
 ^ Pan, Victor; Reif, John (1985), Harvard University Center for Research in Computing Technology Report TR0285, Cambridge, MA: Aiken Computation Laboratory
 ^ "The Inversion of Large Matrices". Byte Magazine. 11 (4): 181–190. April 1986.
 ^ A proof can be found in the Appendix B of Kondratyuk, L. A.; Krivoruchenko, M. I. (1992). "Superconducting quark matter in SU(2) color group". Zeitschrift für Physik A. 344 (1): 99–115. Bibcode:1992ZPhyA.344...99K. doi:10.1007/BF01291027. S2CID 120467300.
 ^ Strang, Gilbert (2003). Introduction to linear algebra (3rd ed.). SIAM. p. 71. ISBN 9780961408893., Chapter 2, page 71
 ^ TzonTzer, Lu; ShengHua, Shiou (2002). "Inverses of 2 × 2 block matrices". Computers & Mathematics with Applications. 43 (1–2): 119–129. doi:10.1016/S08981221(01)002784.
 ^ Bernstein, Dennis (2005). Matrix Mathematics. Princeton University Press. p. 44. ISBN 9780691118024.
 ^ Bernstein, Dennis (2005). Matrix Mathematics. Princeton University Press. p. 45. ISBN 9780691118024.
 ^ T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms, 3rd ed., MIT Press, Cambridge, MA, 2009, §28.2.
 ^ Ran Raz. On the complexity of matrix product. In Proceedings of the thirtyfourth annual ACM symposium on Theory of computing. ACM Press, 2002. doi:10.1145/509907.509932.
 ^ Stewart, Gilbert (1998). Matrix Algorithms: Basic decompositions. SIAM. p. 55. ISBN 9780898714142.
 ^ Haramoto, H.; Matsumoto, M. (2009). "A padic algorithm for computing the inverse of integer matrices". Journal of Computational and Applied Mathematics. 225 (1): 320–322. Bibcode:2009JCoAM.225..320H. doi:10.1016/j.cam.2008.07.044.
 ^ "IML  Integer Matrix Library". cs.uwaterloo.ca. Retrieved 14 April 2018.
 ^ Magnus, Jan R.; Neudecker, Heinz (1999). Matrix Differential Calculus : with Applications in Statistics and Econometrics (Revised ed.). New York: John Wiley & Sons. pp. 151–152. ISBN 047198633X.
 ^ Roman, Stephen (2008), Advanced Linear Algebra, Graduate Texts in Mathematics (Third ed.), Springer, p. 446, ISBN 9780387728285.
 ^ Lin, Lin; Lu, Jianfeng; Ying, Lexing; Car, Roberto; E, Weinan (2009). "Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems". Communications in Mathematical Sciences. 7 (3): 755–777. doi:10.4310/CMS.2009.v7.n3.a12.
Further reading[edit]
 "Inversion of a matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001) [1990]. "28.4: Inverting matrices". Introduction to Algorithms (2nd ed.). MIT Press and McGrawHill. pp. 755–760. ISBN 0262032937.
 Bernstein, Dennis S. (2009). Matrix Mathematics: Theory, Facts, and Formulas (2nd ed.). Princeton University Press. ISBN 9780691140391 – via Google Books.
 Petersen, Kaare Brandt; Pedersen, Michael Syskind (November 15, 2012). "The Matrix Cookbook" (PDF). pp. 17–23.
External links[edit]
This article's use of external links may not follow Wikipedia's policies or guidelines. (June 2015) 
 Sanderson, Grant (August 15, 2016). "Inverse Matrices, Column Space and Null Space". Essence of Linear Algebra. Archived from the original on 20211103 – via YouTube.
 Strang, Gilbert. "Linear Algebra Lecture on Inverse Matrices". MIT OpenCourseWare.
 MoorePenrose Inverse Matrix