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 ${\displaystyle A\in R^{n\times n}}$ ${\displaystyle R}$ ${\displaystyle B\in R^{n\times n}}$

${\displaystyle A\cdot B=B\cdot A=I}$

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 . ${\displaystyle I}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A^{-1}}$ ${\displaystyle R}$ ${\displaystyle B\cdot A=I}$ ${\displaystyle A\cdot B=I}$

Examples

The real matrix

${\displaystyle A={\begin{pmatrix}2&3\\1&2\end{pmatrix}}}$

is regular because it has the inverse

${\displaystyle B={\begin{pmatrix}2&-3\\-1&2\end{pmatrix}}}$,

With

${\displaystyle A\cdot B={\begin{pmatrix}2&3\\1&2\end{pmatrix}}\cdot {\begin{pmatrix}2&-3\\-1&2\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}=I}$.

The real matrix

${\displaystyle A={\begin{pmatrix}2&3\\0&0\end{pmatrix}}}$

is singular, because for any matrix

${\displaystyle B={\begin{pmatrix}a&b\\c&d\end{pmatrix}}}$

applies

${\displaystyle A\cdot B={\begin{pmatrix}2&3\\0&0\end{pmatrix}}\cdot {\begin{pmatrix}a&b\\c&d\end{pmatrix}}={\begin{pmatrix}2a+3c&2b+3d\\0&0\end{pmatrix}}\neq I}$.

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: ${\displaystyle (n\times n)}$ ${\displaystyle A}$ ${\displaystyle K}$

• There is a matrix with . ${\displaystyle B}$ ${\displaystyle AB=I=BA}$
• The determinant of is not zero. ${\displaystyle A}$
• The eigenvalues of are all non-zero. ${\displaystyle A}$
• For all of them there is at least one solution of the linear equation system . ${\displaystyle b\in K^{n}}$ ${\displaystyle x\in K^{n}}$ ${\displaystyle Ax=b}$
• For all of them there is at most one solution of the linear system of equations . ${\displaystyle b\in K^{n}}$ ${\displaystyle x\in K^{n}}$ ${\displaystyle Ax=b}$
• The linear system of equations only has the trivial solution . ${\displaystyle Ax=0}$ ${\displaystyle x=0}$
• The row vectors are linearly independent .
• The row vectors produce . ${\displaystyle K^{n}}$
• The column vectors are linearly independent.
• Create the column vectors . ${\displaystyle K^{n}}$
• By linear described Figure , is injective . ${\displaystyle A}$ ${\displaystyle K^{n}\to K^{n}}$ ${\displaystyle x\mapsto Ax}$
• By linear described Figure , is surjektiv . ${\displaystyle A}$ ${\displaystyle K^{n}\to K^{n}}$ ${\displaystyle x\mapsto Ax}$
• The transposed matrix is invertible. ${\displaystyle A^{T}}$
• The rank of the matrix is the same . ${\displaystyle A}$ ${\displaystyle n}$

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: ${\displaystyle (n\times n)}$ ${\displaystyle A}$ ${\displaystyle R}$

• There is a matrix with . ${\displaystyle B}$ ${\displaystyle AB=I=BA}$
• The determinant of is a unit in (one also speaks of a unimodular matrix ). ${\displaystyle A}$ ${\displaystyle R}$
• There is exactly one solution of the linear equation system for all of them . ${\displaystyle b\in R^{n}}$ ${\displaystyle x\in R^{n}}$ ${\displaystyle Ax=b}$
• For all of them there is at least one solution of the linear equation system . ${\displaystyle b\in R^{n}}$ ${\displaystyle x\in R^{n}}$ ${\displaystyle Ax=b}$
• The row vectors form a basis of . ${\displaystyle R^{n}}$
• Generate the row vectors . ${\displaystyle R^{n}}$
• The column vectors form a basis of . ${\displaystyle R^{n}}$
• Create the column vectors . ${\displaystyle R^{n}}$
• By linear described Figure , is a function onto (or even bijective ). ${\displaystyle A}$ ${\displaystyle R^{n}\to R^{n}}$ ${\displaystyle x\mapsto Ax}$
• The transposed matrix is invertible. ${\displaystyle A^{T}}$

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. ${\displaystyle \mathbb {Z} \to \mathbb {Z} }$ ${\displaystyle x\mapsto 2x}$

Further examples

The matrix

${\displaystyle A={\begin{pmatrix}3x^{3}&x^{2}-1\\3x^{2}+3&x\end{pmatrix}}}$

with entries from the polynomial ring has the determinant and can be inverted into . So is regular in ; the inverse is ${\displaystyle R=\mathbb {R} [x]}$ ${\displaystyle \det A=3}$ ${\displaystyle 3}$ ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle R^{2\times 2}}$

${\displaystyle B={\frac {1}{3}}{\begin{pmatrix}x&1-x^{2}\\-3x^{2}-3&3x^{3}\end{pmatrix}}}$.

The matrix

${\displaystyle A={\begin{pmatrix}[3]&[7]\\\left[1\right]&[9]\end{pmatrix}}}$

with entries from the remainder class ring has the determinant . Since and not prime are is in not invertible. Hence it is not regular. ${\displaystyle \mathbb {Z} /12\mathbb {Z} }$ ${\displaystyle \det A=[20]=[8]}$ ${\displaystyle 8}$ ${\displaystyle 12}$ ${\displaystyle \det A}$ ${\displaystyle \mathbb {Z} /12\mathbb {Z} }$ ${\displaystyle A}$

properties

If the matrix is regular, then is also regular with the inverse ${\displaystyle A}$ ${\displaystyle A^{-1}}$

${\displaystyle \left(A^{-1}\right)^{-1}=A}$.

If the two matrices and are regular, then their product is also regular with the inverse ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\cdot B}$

${\displaystyle \left(A\cdot B\right)^{-1}=B^{-1}\cdot A^{-1}}$.

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 ${\displaystyle \operatorname {GL} (n,R)}$ ${\displaystyle A}$

${\displaystyle A\cdot B=A\cdot C\Rightarrow B=C}$

and

${\displaystyle B\cdot A=C\cdot A\Rightarrow B=C}$,

where and are arbitrary matrices of suitable size. ${\displaystyle B}$ ${\displaystyle C}$

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)