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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
by adding the corresponding entries together. However, there are other operations which could also be considered
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

for matrices, such as the
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
and the
Kronecker sum In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation (mathematics), operation on two matrix (mathematics), matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is ...
.

# Entrywise sum

Two matrices must have an equal number of rows and columns to be added. In which case, the sum of two matrices A and B will be a matrix which has the same number of rows and columns as A and B. The sum of A and B, denoted , is computed by adding corresponding elements of A and B: :$\begin \mathbf+\mathbf & = \begin a_ & a_ & \cdots & a_ \\ a_ & a_ & \cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ & a_ & \cdots & a_ \\ \end + \begin b_ & b_ & \cdots & b_ \\ b_ & b_ & \cdots & b_ \\ \vdots & \vdots & \ddots & \vdots \\ b_ & b_ & \cdots & b_ \\ \end \\ & = \begin a_ + b_ & a_ + b_ & \cdots & a_ + b_ \\ a_ + b_ & a_ + b_ & \cdots & a_ + b_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ + b_ & a_ + b_ & \cdots & a_ + b_ \\ \end \\ \end\,\!$ Or more concisely (assuming that ): :$c_=a_+b_$ For example: :$\begin 1 & 3 \\ 1 & 0 \\ 1 & 2 \end + \begin 0 & 0 \\ 7 & 5 \\ 2 & 1 \end = \begin 1+0 & 3+0 \\ 1+7 & 0+5 \\ 1+2 & 2+1 \end = \begin 1 & 3 \\ 8 & 5 \\ 3 & 3 \end$ Similarly, it is also possible to subtract one matrix from another, as long as they have the same dimensions. The difference of A and B, denoted , is computed by subtracting elements of B from corresponding elements of A, and has the same dimensions as A and B. For example: :$\begin 1 & 3 \\ 1 & 0 \\ 1 & 2 \end - \begin 0 & 0 \\ 7 & 5 \\ 2 & 1 \end = \begin 1-0 & 3-0 \\ 1-7 & 0-5 \\ 1-2 & 2-1 \end = \begin 1 & 3 \\ -6 & -5 \\ -1 & 1 \end$

# Direct sum

Another operation, which is used less often, is the direct sum (denoted by ⊕). Note the Kronecker sum is also denoted ⊕; the context should make the usage clear. The direct sum of any pair of matrices A of size ''m'' × ''n'' and B of size ''p'' × ''q'' is a matrix of size (''m'' + ''p'') × (''n'' + ''q'') defined as: :$\mathbf \oplus \mathbf = \begin \mathbf & \boldsymbol \\ \boldsymbol & \mathbf \end = \begin a_ & \cdots & a_ & 0 & \cdots & 0 \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_ & \cdots & a_ & 0 & \cdots & 0 \\ 0 & \cdots & 0 & b_ & \cdots & b_ \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & b_ & \cdots & b_ \end$ For instance, :$\begin 1 & 3 & 2 \\ 2 & 3 & 1 \end \oplus \begin 1 & 6 \\ 0 & 1 \end = \begin 1 & 3 & 2 & 0 & 0 \\ 2 & 3 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 6 \\ 0 & 0 & 0 & 0 & 1 \end$ The direct sum of matrices is a special type of
block matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. In particular, the direct sum of square matrices is a
block diagonal matrix In mathematics, a block matrix or a partitioned matrix is a matrix (mathematics), matrix that is ''Interpretation (logic), interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block m ...
. The
adjacency matrix In graph theory In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theor ...
of the union of disjoint
graphs Graph may refer to: Mathematics *Graph (discrete mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes an ...
(or
multigraph In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s) is the direct sum of their adjacency matrices. Any element in the
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
of two
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s of matrices can be represented as a direct sum of two matrices. In general, the direct sum of ''n'' matrices is: :$\bigoplus_^ \mathbf_ = \operatorname\left( \mathbf_1, \mathbf_2, \mathbf_3, \ldots, \mathbf_n\right) = \begin \mathbf_1 & \boldsymbol & \cdots & \boldsymbol \\ \boldsymbol & \mathbf_2 & \cdots & \boldsymbol \\ \vdots & \vdots & \ddots & \vdots \\ \boldsymbol & \boldsymbol & \cdots & \mathbf_n \\ \end\,\!$ where the zeros are actually blocks of zeros (i.e., zero matrices).

# Kronecker sum

The Kronecker sum is different from the direct sum, but is also denoted by ⊕. It is defined using the
Kronecker product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
⊗ and normal matrix addition. If A is ''n''-by-''n'', B is ''m''-by-''m'' and $\mathbf_k$ denotes the ''k''-by-''k''
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

then the Kronecker sum is defined by: :$\mathbf \oplus \mathbf = \mathbf \otimes \mathbf_m + \mathbf_n \otimes \mathbf.$

*
Matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

*
Vector addition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

* *