Regular matrix
A regular , invertible, or nonsingular matrix is a square matrix in mathematics that has an inverse . Regular matrices can be characterized in several equivalent ways. For example, regular matrices are characterized by the fact that the linear mapping they describe is bijective . Therefore, a linear system of equations with a regular coefficient matrix can always be solved uniquely. The set of regular fixed-size matrices with entries from a ring or body forms with the Matrix multiplication as a link the general linear group .
Not every square matrix has an inverse. A square matrix that has no inverse is called a singular matrix .
definition
A square matrix with entries from a unitary ring (in practice mostly the field of real numbers ) is called regular if another matrix exists such that
holds, where denotes the identity matrix . The matrix here is uniquely determined and is called inverse matrix to . The inverse of a matrix is usually denoted by. If a commutative ring , body or inclined body , then the two conditions are equivalent, that is, a left-inverse matrix is then also right-inverse and vice versa, i.e. the above condition can be weakened by or .
Examples
The real matrix
is regular because it has the inverse
- ,
With
- .
The real matrix
is singular, because for any matrix
applies
- .
Equivalent characterizations
Regular matrices over a body
A matrix with entries from a field , for example the real or complex numbers , can be inverted if and only if one of the following equivalent conditions is met:
- There is a matrix with .
- The determinant of is not zero.
- The eigenvalues of are all non-zero.
- For all of them there is at least one solution of the linear equation system .
- For all of them there is at most one solution of the linear system of equations .
- The linear system of equations only has the trivial solution .
- The row vectors are linearly independent .
- The row vectors produce .
- The column vectors are linearly independent.
- Create the column vectors .
- By linear described Figure , is injective .
- By linear described Figure , is surjektiv .
- The transposed matrix is invertible.
- The rank of the matrix is the same .
Regular matrices over a unitary commutative ring
More generally, a matrix with entries from a commutative ring with one can be inverted if and only if one of the following equivalent conditions is met:
- There is a matrix with .
- The determinant of is a unit in (one also speaks of a unimodular matrix ).
- There is exactly one solution of the linear equation system for all of them .
- For all of them there is at least one solution of the linear equation system .
- The row vectors form a basis of .
- Generate the row vectors .
- The column vectors form a basis of .
- Create the column vectors .
- By linear described Figure , is a function onto (or even bijective ).
- The transposed matrix is invertible.
The main difference from the case of a body is thus here that, in general from the injectivity of a linear map no longer their surjectivity (and thus their bijectivity) follows, as the simple example , shows.
Further examples
The matrix
with entries from the polynomial ring has the determinant and can be inverted into . So is regular in ; the inverse is
- .
The matrix
with entries from the remainder class ring has the determinant . Since and not prime are is in not invertible. Hence it is not regular.
properties
If the matrix is regular, then is also regular with the inverse
- .
If the two matrices and are regular, then their product is also regular with the inverse
- .
The set of regular matrices of a fixed size thus forms a (generally non-commutative ) group , the general linear group, with the matrix multiplication as a link . In this group, the identity matrix is the neutral element and the inverse matrix is the inverse element . The truncation rules therefore also apply to a regular matrix
and
- ,
where and are arbitrary matrices of suitable size.
literature
- Peter Knabner , Wolf Barth : Lineare Algebra. Basics and Applications. Springer Spectrum, Berlin / Heidelberg 2013, ISBN 978-3-642-32185-6 .
Web links
- Non-singular matrix . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- Eric W. Weisstein : Nonsingular Matrix . In: MathWorld (English).
- CWoo: Invertible matrix . In: PlanetMath . (English)