Matrix Addition

The content in today's blog is taken straight from Gareth Williams' Linear Algebra.

Matrices in terms of their properties present an interesting contrast to numbers. Whereas with numbers, we can perform addition and subtraction on all elements, with matrices, this is not the case.

Indeed, one of the most important properties of matrices is that there are certain preconditions required before operations can be performed on two matrices. For example, addition can only occur between two matrices that have the same dimensions. It is not possible to add a 1 x 2 matrix with a 2 x 2 matrix.

Definition 1. Zero Matrix: Z_m,n

A zero matrix is any matrix which consists completely of 0's.

For example, below is Z_2,3:



Below are the basic properties of matrix addition.

Property 1: Commutative property of Addition

A + B = B + A

where A and B are matrices of the same dimension and consist of scalar values.

Proof:

(1) Let A = set of values a_i,j where i is the row and j is the column.

(2) Let B = set of values b_i,j

(3) A + B = the set of values a_i,j + b_i,j

(4) B + A = the set of values b_i,j + a_i,j

(5) Clearly, since all a_i,j and b_i,j are scalar, a_i,j + b_i,j = b_i,j + a_i,j

QED

Property 2: Associative property of Addition

A + (B + C) = (A + B) + C

where A, B, and C are matrices of the same dimension and consist of scalar values.

Proof:

(1) All A,B,C have the same dimensions since this is a prerequisite for addition.

(2) Let A = the set a_i,j, B = the set b_i,j, C = the set c_i,j

(3) A + (B + C) = the set a_i,j + (b_i,j + c_i,j)

(4) (A + B) + C = the set (a_i,j + b_i,j) + c_i,j

(5) Since a_i,j, b_i,j, c_i,j are all scalar, a_i,j + (b_i,j + c_i,j) = (a_i,j + b_i,j) + c_i,j

QED

Property 3: Addition by Z_m,n

A + Z_m,n = Z_m,n+ A = A

where A is an m x n matrix which consists of scalar values.

Proof:

(1) Let A be the set of a_i,j

(2) Let Z be the set of z_i,j where each z_i,j=0 [See Definition 1 above]

(3) A + Z_m,n = the set of a_i,j + z_i,j = the set of a_i,j = A

(4) Z_m,n + A = the set of z_i,j + a_i,j = a_i,j = A.

QED

Property 4: Distributive Property of Scalars with Addition of Matrices

c(A + B) = cA + cB

where A and B are matrices of the same dimension and consist of scalar values and c is a scalar value.

Proof:

(1) Let A = the set of a_i,j, let B = the set of b_i,j

(2) A + B = the set a_i,j + b_i,j

(3) c(A+B) = the set c(a_i,j + b_i,j) = a_i,jc + b_i,jc .

(4) cA + cB = the set of a_i,jc + the set of b_i,jc = the set of a_i,jc + b_i,jc.

QED

Property 5: Distributive Property of Matrices with Addition of Scalars

(a + b)C = aC + bC

where a,b are scalar values and C is a matrix of scalar values.

Proof:

(1) Let C = the set of c_i,j

(2) (a + b)C = the set of (a + b)c_i,j = ac_i,j + bc_i,j

(3) aC + bC = the set of ac_i,j+ the set of bc_i,j = the set of ac_i,j + bc_i,j

QED

Property 6: Associative Property of Multiplication of Scalars with Matrices

(ab)C = a(bC)

where a,b are scalar values and C is a matrix of scalar values.

Proof:

(1) Let C be the set of c_i,j

(2) (ab)C = the set of abc_i,j

(3) a(bC) = a*(the set of bc_i,j) = the set of abc_i,j

QED

References

• Gareth Williams, Linear Algebra with Applications, Wm. C. Brown Publishers, 1996.