Matrix addition

The Matrizenaddtion or matrix addition is in mathematics an additive combination of two matrices of the same size. The result is called a Matrizenaddtion Matrizensumme, matrix sum or sum matrix and is obtained by component-wise addition of the corresponding entries in each of the two output matrices. The Matrizenaddtion is associative, commutative and distributive with the matrix multiplication.

The set of matrices forms the matrix addition an additive group, the neutral element is the zero matrix. The amount of square matrices over a ring forms with the matrix addition and matrix multiplication turn a ring. The set of matrices over a field forms with the matrix addition and scalar multiplication of a vector space.

Definition

And is a ring, and two matrices over, then the Matrizensumme of and

Defined. The sum matrix thus results by component-wise addition of the corresponding entries of the two output matrices. It is only defined for the case, that the two output matrices of the same size. The resulting matrix has then also this size.

Example

The Matrizensumme the two real (2 × 2) - matrices

Is obtained as

Properties

The matrix addition inherits the properties of the underlying ring. It is associative, ie for matrices applies

And commutative, ie

Further, the matrix addition is compatible with the multiplication of scalars, ie

Together with the matrix multiplication also apply the distributive

Next is true for the matrix transpose of a sum of two matrices

The sum of two symmetric matrices is therefore symmetrical again.

Algebraic Structures

Matrices as a group

The set of matrices of fixed size forms with the matrix addition an additive group. The neutral element in this group is the zero matrix in which all entries are equal to the zero element. Thus true for all matrices

The additive inverse to a matrix element is then the matrix

Wherein the additive inverse element in representing. The difference between two matrices is thus given by

Matrix rings

The amount of square matrices of fixed size forms with the matrix addition and matrix multiplication a ( non-commutative ) ring. Is the underlying ring is unitary, then the corresponding die ring is unitary, with the fuel element is shown by the unit matrix.

Also a ring is the set of all matrices of any fixed size of the matrix addition and with the Hadamard product. Is unitary, then this matrix ring also has a unit element, the fuel matrix in which all elements are equal to the identity element of the output ring.

Die space

The set of matrices of arbitrary fixed size over a field forms with the matrix addition and scalar multiplication of a vector space. The default base for these die space consists of the set of standard matrices, where the entry at the location is one and all other entries are zero. The die cavity therefore has the dimension.

If a matrix over the field of real or complex numbers, and a matrix norm, then applies, by definition, a norm, the triangle inequality

The standard is therefore a Matrizensumme most as large as the sum of the norms of the addends.