# 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}}$

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

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

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

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