Invertible matrix

The regular, invertible or non-singular matrix is a term from the mathematical subfield of linear algebra. A square matrix can be inverted, if there is a further matrix, so that

Holds, where the identity matrix. In this case also applies

The matrix here is uniquely determined and is called the inverse matrix or just shortly inverse. One typically writes   for the inverse matrix.

Invertible matrices are characterized by the fact that the described by them linear mapping is bijective. This leads to a linear equation system with an invertible matrix of coefficients is a unique solution.

Not every square matrix has an inverse exists. A matrix has no inverse is called the singular matrix. The set of all invertible matrices over a base ( or base ring ) forms a group under matrix multiplication, the general linear group.

A generalization of the inverse matrix of singular and non- square matrices is enabled by the concept of pseudo-inverse.

  • 2.1 Gauss-Jordan algorithm
  • 2.2 adjuncts 2.2.1 Formula for 2x2 matrices
  • 2.2.2 Formula for 3x3 matrices
  • 2.2.3 Formula for 4x4 matrices
  • 2.2.4 Derivation of the formula

Mathematical definition

Invertible matrices over a commutative unitary ring

It is a commutative ring with identity, and is a matrix with entries from.

The matrix is ​​invertible if one of the following equivalent conditions is satisfied:

  • There is a matrix (the unit matrix ).
  • The determinant of is a unit in ( one also speaks of a unimodular matrix).
  • For all there exists a unique solution of the linear system of equations.
  • The row vectors form a basis of.
  • The column vectors form a basis of.
  • The by - described linear map is surjective ( or even bijective ).
  • The transposed matrix is invertible.
  • For all there exists at least one solution of the linear system of equations.
  • Create the row vectors.
  • Create the column vectors.

Invertible matrices over a field

It was a body ( for example, or ) and is a matrix with entries from the body. Since every body is a ring, all of the above statements are also valid here, but can be very much easier to express and additionally transform yet to equivalent statements.

Then is invertible if one of the following equivalent conditions is satisfied:

  • There is a matrix (the unit matrix ).
  • The determinant is non-zero.
  • Is not an eigenvalue of.
  • For all there exists at least one solution of the linear system of equations.
  • For all there exists at most one solution to the linear system.
  • The linear system of equations has only the trivial solution
  • The row vectors are linearly independent.
  • Create the row vectors.
  • The column vectors are linearly independent.
  • Create the column vectors.
  • The described by linear mapping is injective.
  • The described by linear mapping is surjective.
  • The transposed matrix is invertible.
  • The rank of the matrix is the same.

Calculation of the inverse of a matrix

To calculate the inverse are two options available: the Gauss-Jordan algorithm and the adjoint. In particular, by means of the adjoint can in principle formulas for matrices with specified rank derived. However, these are too extensive to be used efficiently, so sometimes the formulas listed below are used only for 2x2 and 3x3 matrices. In practical applications, as far as possible to dispense with the calculation of the inverse matrix and, instead, dissolved, a system of linear equations.

To test the quality of the numerical algorithms for inverting matrices using the Hilbert matrix, since it is relatively ill-conditioned.

Gauss-Jordan algorithm

The inverse matrix can be calculated from the formula. To form the matrix is ​​and applies this to the Gauss-Jordan algorithm. After implementation of the algorithm has a block matrix from which one can read off directly.

Example:

Wanted is the inverse of the matrix

The block matrix is

Applying the Gauss-Jordan algorithm leads to the matrix

From this, the inverse matrix directly read:

Adjuncts

By means of the adjoint and the determinant of a matrix, whose inverse is calculated using the following formula:

This statement is for - and - matrices from the following formulas:

Formula for 2x2 matrices

Formula for 3x3 matrices

Formula for 4x4 matrices

The calculation rule for 4x4 matrices is more complicated: external link

Derivation of the formula

The idea to calculate the inverse of a matrix using the adjuncts, derived directly from Cramer's rule. After this can be the system of equations

By using the- th unit vector on the right side

Solve. The matrix is formed from by replacing the th column by the - th unit vector. Whose determinant is identical due to the simple shape of the unit vector with the cofactor. It can be seen that correspond to the columns of the inverse matrix to. To this end, multiply both sides of the equation system shown initially from the right with the transposed - th unit vector and is the sum over all.

Approximation

Has a matrix property

Then it is regular and its inverse may be expressed by the following geometric series:

Special classes of matrices

There are some classes of matrices that are based on their structure very easy to invert. These include the diagonal and triangular matrices. Explicit representations of the inverses exist for unitary matrices, and for changes of rank 1 ( Sherman - Morrison - Woodbury formula).

Properties

Is an eigenvalue of the regular matrix with eigenvector, so eigenvalue of the inverse matrix is also the eigenvector.

Calculation rules

The product of two invertible matrices is invertible again. It is

The inverse of the transposed matrix A corresponds to the transpose of the inverse matrix:

The inverse of a matrix is also invertible. The inverse of the inverse is just again the matrix itself:

The inverse of a matrix multiplied by a scalar

Unit (ring theory) Hilbert-Matrix Adjugate matrix Triangular matrix Sherman–Morrison formula Wolfram Research
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