Inverse element - Citizendia

In mathematics, the idea of inverse element generalises the concepts of negation, in relation to addition, and reciprocal, in relation to multiplication. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In mathematics the additive inverse, or opposite, of a Number n is the number that when added to n, yields zero. Addition is the mathematical process of putting things together In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which The intuition is of an element that can 'undo' the effect of combination with another given element.

1 Formal definition
2 Calculation
3 Example
4 See also
5 References

Formal definition

Let $S$ be a set with a binary operation $*$ . In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two If $e$ is an identity element of $(S, * )$ and $a * b = e$ , then $a$ is called a left inverse of $b$ and $b$ is called a right inverse of $a$ . In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that If an element $x$ is both a left inverse and a right inverse of $y$ , then $x$ is called a two-sided inverse, or simply an inverse, of $y$ . An element with a two-sided inverse in $S$ is called invertible in $S$ . An element with an inverse element only on one side is left invertible, resp. right invertible.

Just like $(S, * )$ can have several left identities or several right identities, it is possible for an element to have several left inverses or several right inverses (but note that their definition above uses a two-sided identity $e$ ). It can even have several left inverses and several right inverses.

If the operation $*$ is associative then if an element has both a left inverse and a right inverse, they are equal and unique. In Mathematics, associativity is a property that a Binary operation can have In this case, the set of (left and right) invertible elements is a group, called the group of units of $S$ , and denoted by $U (S)$ or $S *$ . In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i

Calculation

Every real number $x$ has an additive inverse (i. In Mathematics, the real numbers may be described informally in several different ways In mathematics the additive inverse, or opposite, of a Number n is the number that when added to n, yields zero. e. an inverse with respect to addition) given by $- x$ . Addition is the mathematical process of putting things together Every nonzero real number $x$ has a multiplicative inverse (i. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which e. an inverse with respect to multiplication) given by $\frac 1{x}$ . By contrast, zero has no multiplicative inverse.

A function $g$ is the left (resp. right) inverse of a function $f$ (for function composition), if and only if $g o f$ (resp. In Mathematics, a composite function represents the application of one function to the results of another $f o g$ ) is the identity function on the domain (resp. In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined codomain) of $f$ . In Mathematics, the codomain, or target, of a function f: X → Y is the set

A square matrix $M$ with entries in a field $K$ is invertible (in the set of all square matrices of the same size, under matrix multiplication) if and only if its determinant is different from zero. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n If the determinant of $M$ is zero, it is impossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the other one. See invertible matrix for more. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by-

More generally, a square matrix over a commutative ring $R$ is invertible if and only if its determinant is invertible in $R$ . In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property ↔

Non-square matrices of full rank have one-sided inverses:^[1]

For $A:m\times n \mid m>n$ we have a left inverse: $\underbrace{ (A^{T}A)^{-1}A^{T} }_{ A^{-1}_\text{left} } A = I_{n}$
For $A:m\times n \mid m<n$ we have a right inverse: $A \underbrace{ A^{T}(AA^{T})^{-1} }_{ A^{-1}_\text{right} } = I_{m}$

No rank-deficient matrix has any (even one-sided) inverse. The column rank of a matrix A is the maximal number of Linearly independent columns of A. However, the Moore-Penrose pseudoinverse exists for all matrices, and coincides with the left or right (or true) inverse when it exists. In Mathematics, and in particular Linear algebra, the pseudoinverse A^+ of an m \times n matrix A is a generalization

Example

$A:2\times 3 = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$
So, as m<n, we have a right inverse. $A^{-1}_{right} = A^{T}(AA^{T})^{-1}$
$AA^{T} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\cdot \begin{bmatrix} 1 & 4\\ 2 & 5\\ 3 & 6 \end{bmatrix} = \begin{bmatrix} 14 & 32\\ 32 & 77 \end{bmatrix}$
$(AA^{T})^{-1} = \begin{bmatrix} 14 & 32\\ 32 & 77 \end{bmatrix}^{-1} = \frac{1}{54} \begin{bmatrix} 77 & -32\\ -32 & 14 \end{bmatrix}$

$A^{T}(AA^{T})^{-1} = \frac{1}{54}\begin{bmatrix} 1 & 4\\ 2 & 5\\ 3 & 6 \end{bmatrix} \cdot \begin{bmatrix} 77 & -32\\ -32 & 14 \end{bmatrix} = \frac{1}{18} \begin{bmatrix} -17 & 8\\ -2 & 2\\ 13 & -4 \end{bmatrix} = A^{-1}_{right}$

The left inverse doesn't exist, because $A^{T}A = \begin{bmatrix} 1 & 4\\ 2 & 5\\ 3 & 6 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} 17 & 22 & 27 \\ 22 & 29 & 36\\ 27 & 36 & 45 \end{bmatrix}$ Is a singular matrix, and can't be inverted.

References

^ MIT Professor Gilbert Strang Linear Algebra Lecture #33 - Left and Right Inverses; Pseudoinverse.

In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i

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Contents

Formal definition

Calculation

Example

See also

References