In

Abstract nonsense: Direct Sum of Linear Transformations and Direct Sum of Matrices

Matrix Algebra and R

Linear algebra Bilinear operators

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 ...

, matrix addition is the operation of adding two 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: :$\backslash begin\; \backslash mathbf+\backslash mathbf\; \&\; =\; \backslash begin\; a\_\; \&\; a\_\; \&\; \backslash cdots\; \&\; a\_\; \backslash \backslash \; a\_\; \&\; a\_\; \&\; \backslash cdots\; \&\; a\_\; \backslash \backslash \; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; a\_\; \&\; a\_\; \&\; \backslash cdots\; \&\; a\_\; \backslash \backslash \; \backslash end\; +\; \backslash begin\; b\_\; \&\; b\_\; \&\; \backslash cdots\; \&\; b\_\; \backslash \backslash \; b\_\; \&\; b\_\; \&\; \backslash cdots\; \&\; b\_\; \backslash \backslash \; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; b\_\; \&\; b\_\; \&\; \backslash cdots\; \&\; b\_\; \backslash \backslash \; \backslash end\; \backslash \backslash \; \&\; =\; \backslash begin\; a\_\; +\; b\_\; \&\; a\_\; +\; b\_\; \&\; \backslash cdots\; \&\; a\_\; +\; b\_\; \backslash \backslash \; a\_\; +\; b\_\; \&\; a\_\; +\; b\_\; \&\; \backslash cdots\; \&\; a\_\; +\; b\_\; \backslash \backslash \; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; a\_\; +\; b\_\; \&\; a\_\; +\; b\_\; \&\; \backslash cdots\; \&\; a\_\; +\; b\_\; \backslash \backslash \; \backslash end\; \backslash \backslash \; \backslash end\backslash ,\backslash !$ Or more concisely (assuming that ): :$c\_=a\_+b\_$ For example: :$\backslash begin\; 1\; \&\; 3\; \backslash \backslash \; 1\; \&\; 0\; \backslash \backslash \; 1\; \&\; 2\; \backslash end\; +\; \backslash begin\; 0\; \&\; 0\; \backslash \backslash \; 7\; \&\; 5\; \backslash \backslash \; 2\; \&\; 1\; \backslash end\; =\; \backslash begin\; 1+0\; \&\; 3+0\; \backslash \backslash \; 1+7\; \&\; 0+5\; \backslash \backslash \; 1+2\; \&\; 2+1\; \backslash end\; =\; \backslash begin\; 1\; \&\; 3\; \backslash \backslash \; 8\; \&\; 5\; \backslash \backslash \; 3\; \&\; 3\; \backslash end$ 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: :$\backslash begin\; 1\; \&\; 3\; \backslash \backslash \; 1\; \&\; 0\; \backslash \backslash \; 1\; \&\; 2\; \backslash end\; -\; \backslash begin\; 0\; \&\; 0\; \backslash \backslash \; 7\; \&\; 5\; \backslash \backslash \; 2\; \&\; 1\; \backslash end\; =\; \backslash begin\; 1-0\; \&\; 3-0\; \backslash \backslash \; 1-7\; \&\; 0-5\; \backslash \backslash \; 1-2\; \&\; 2-1\; \backslash end\; =\; \backslash begin\; 1\; \&\; 3\; \backslash \backslash \; -6\; \&\; -5\; \backslash \backslash \; -1\; \&\; 1\; \backslash end$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: :$\backslash mathbf\; \backslash oplus\; \backslash mathbf\; =\; \backslash begin\; \backslash mathbf\; \&\; \backslash boldsymbol\; \backslash \backslash \; \backslash boldsymbol\; \&\; \backslash mathbf\; \backslash end\; =\; \backslash begin\; a\_\; \&\; \backslash cdots\; \&\; a\_\; \&\; 0\; \&\; \backslash cdots\; \&\; 0\; \backslash \backslash \; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; a\_\; \&\; \backslash cdots\; \&\; a\_\; \&\; 0\; \&\; \backslash cdots\; \&\; 0\; \backslash \backslash \; 0\; \&\; \backslash cdots\; \&\; 0\; \&\; b\_\; \&\; \backslash cdots\; \&\; b\_\; \backslash \backslash \; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; 0\; \&\; \backslash cdots\; \&\; 0\; \&\; b\_\; \&\; \backslash cdots\; \&\; b\_\; \backslash end$ For instance, :$\backslash begin\; 1\; \&\; 3\; \&\; 2\; \backslash \backslash \; 2\; \&\; 3\; \&\; 1\; \backslash end\; \backslash oplus\; \backslash begin\; 1\; \&\; 6\; \backslash \backslash \; 0\; \&\; 1\; \backslash end\; =\; \backslash begin\; 1\; \&\; 3\; \&\; 2\; \&\; 0\; \&\; 0\; \backslash \backslash \; 2\; \&\; 3\; \&\; 1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 1\; \&\; 6\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 0\; \&\; 1\; \backslash end$ The direct sum of matrices is a special type ofblock 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:
:$\backslash bigoplus\_^\; \backslash mathbf\_\; =\; \backslash operatorname(\; \backslash mathbf\_1,\; \backslash mathbf\_2,\; \backslash mathbf\_3,\; \backslash ldots,\; \backslash mathbf\_n)\; =\; \backslash begin\; \backslash mathbf\_1\; \&\; \backslash boldsymbol\; \&\; \backslash cdots\; \&\; \backslash boldsymbol\; \backslash \backslash \; \backslash boldsymbol\; \&\; \backslash mathbf\_2\; \&\; \backslash cdots\; \&\; \backslash boldsymbol\; \backslash \backslash \; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; \backslash boldsymbol\; \&\; \backslash boldsymbol\; \&\; \backslash cdots\; \&\; \backslash mathbf\_n\; \backslash \backslash \; \backslash end\backslash ,\backslash !$
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 theKronecker 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 $\backslash mathbf\_k$ 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:
:$\backslash mathbf\; \backslash oplus\; \backslash mathbf\; =\; \backslash mathbf\; \backslash otimes\; \backslash mathbf\_m\; +\; \backslash mathbf\_n\; \backslash otimes\; \backslash mathbf.$
See also

*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 ...

Notes

References

* *External links

*{{PlanetMath , urlname=DirectSumOfMatrices , title= Direct sum of matricesAbstract nonsense: Direct Sum of Linear Transformations and Direct Sum of Matrices

Matrix Algebra and R

Linear algebra Bilinear operators