In linear algebra, an invertible matrix (non-singular, non-degenerate or regular) 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.
Basic idea
A matrix can be viewed as a rule for transforming vectors. For example, a real matrix defines a linear transformation
from the set of -tuples of real numbers to itself. The matrix is invertible when this transformation can be undone by another linear transformation. In that case, there is a matrix such that applying and then , or applying and then , returns every vector to where it started.
Geometrically, an invertible matrix does not collapse space into a lower-dimensional set. It sends distinct vectors to distinct vectors and reaches every vector in the target space. For a real matrix, this is reflected by its determinant: an invertible matrix has nonzero determinant, while a matrix with determinant zero collapses volume to zero and is not invertible.
Algebraically, invertibility means that the linear system
has a unique solution for every vector . Equivalently, the columns of form a basis of the vector space. These and several other equivalent characterizations are summarized by the invertible matrix theorem.
Examples An invertible matrix 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.
A non-invertible matrix This is a non-invertible (singular) matrix:
This matrix is not invertible, because different vectors and have the same value of , for example, and both give . So it is impossible to reverse the transformation , since different inputs produce the same output. The determinant of is 0, which is a necessary and sufficient condition for a matrix to be non-invertible.
Explicit formula for 2x2 matrices For an arbitrary 2-by-2 matrix defined as , its inverse is .
Criteria for invertibility A matrix (over a field such as the real numbers) is invertible if and only if the only solution to the equation is the zero vector. That is, the nullity of is zero: its nullspace consists only of the zero vector. Equivalently, an matrix is invertible if its rank is , by the rank-nullity theorem. Such a matrix is said to be of full rank. Geometrically, this means that the column space of is all of . The rank and nullity characterizations are related in the following way: having full rank means that the matrix maps onto all of (surjectively), and having nullity zero means that the matrix is one-to-one. That is, it is a bijection, and therefore inveritble.
A more computational criterion for invertibility comes from the standard way of determining the rank and nullity of a matrix. For a square matrix, a matrix is invertible if and only if its row-reduced echelon form is the identity matrix. The reason is that this is the only row-reduced echelon matrix having full rank: all ones down the diagonal and zeros everywhere else.
Another criterion is that a matrix is invertible if and only if its determinant is non-zero.
Methods of matrix inversion
There are many methods to compute the inverse matrix, when it exists. A basic method is to use Gaussian elimination, for example. This proceeds by considering the inverse matrix as a product of elementary matrices, where the are the elementary matrices corresponding to the elementary row operations needed to put into its row-reduced echelon form. This method has the advantage that it will produce a row-reduced echelon form regardless of invertibility, and so gives a criterion to decide if a matrix is invertible which is often more efficient than computing the determinant: the matrix is invertible if and only if, at the end of the process, the echelon form is the identity matrix rather than some matrix of lower rank.
To convert this to a practical method for determining using this method, one uses an augmented matrix whose left side is the matrix to invert and whose right side is 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.
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 (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: is invertible, i.e. it has an inverse under matrix multiplication, i.e., there exists a such that . (In that statement, "invertible" can equivalently be replaced with "left-invertible" or "right-invertible" in which one-sided inverses are considered.) The linear transformation mapping to is invertible, i.e., it has an inverse under function composition. (There, again, "invertible" can equivalently be replaced with either "left-invertible" or "right-invertible".) 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 . (There, "bijective" can equivalently be replaced with "injective" or "surjective".) The columns of form a basis of . (In this statement, "basis" can equivalently be replaced with either "linearly independent set" or "spanning set") The rows of form a basis of . (Similarly, here, "basis" can equivalently be replaced with either "linearly independent set" or "spanning set") 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 (i.e. multiplicatively invertible element) of that ring. The number 0 is not an eigenvalue of . (More generally, a number is an eigenvalue of if the matrix is singular, where is the identity matrix.) 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 .
The rows of the inverse matrix of a matrix are orthonormal to the columns of (and vice versa interchanging rows for columns). To see this, suppose that where the rows of are denoted as and the columns of 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 are known. In which case, one can apply the iterative Gram–Schmidt process to this initial set to determine the rows of the inverse .
A matrix that is its own inverse (i.e., a matrix such that and consequently ) is called an involutory matrix.
In relation to its adjugate The adjugate of a matrix is a matrix which exists independently of the invertibility of . It satisfies the identity
In particular, if is invertible, then
In relation to the identity matrix It follows from the associativity of matrix multiplication that if
for finite square matrices and , then also
. This identity does not hold for non-square rectangular matrices, and need not be true for linear operators in infinite dimensions.
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 by
To 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,
The derivative of a matrix determinant with respect to one of its elements is a matrix cofactor, which exhibits the transpose of the matrix inverse:
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 (such as the Moore–Penrose inverse), 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 (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 left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings.