Matrix Subtraction

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.

For a vector, $\vec\!$, adding two matrices would have the geometric effect of applying each matrix transformation separately onto $\vec\!$, then adding the transformed vectors.

:$\mathbf\vec + \mathbf\vec = (\mathbf + \mathbf)\vec\!$

Definition

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:

:$\begin 1 & 3 \\ 1 & 0 \\ 1 & 2 \end - \begin 0 & 0 \\ 7 & 5 \\ 2 & 1 \end = \begin 1-0 & 3-0 \\ 1-7 & 0-5 \\ 1-2 & 2-1 \end = \begin 1 & 3 \\ -6 & -5 \\ -1 & 1 \end$

See also

* Matrix multiplication
* Vector addition
* Direct sum of matrices
* Kronecker sum

Notes

References

*
* {{cite book , last1=Riley , first1=K.F. , last2=Hobson , first2=M.P. , last3=Bence , first3=S.J. , title=Mathematical methods for physics and engineering , edition=3 , publisher=Cambridge University Press , year=2006 , doi=10.1017/CBO9780511810763 , isbn=978-0-521-86153-3 , url=https://archive.org/details/mathematicalmeth00rile , url-access=registration

External links

Matrix Algebra and R

Linear algebra
Bilinear maps