## Monday, April 16, 2007

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: Zm,n

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

For example, below is Z2,3:

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 ai,j where i is the row and j is the column.

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

(3) A + B = the set of values ai,j + bi,j

(4) B + A = the set of values bi,j + ai,j

(5) Clearly, since all ai,j and bi,j are scalar, ai,j + bi,j = bi,j + ai,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 ai,j, B = the set bi,j, C = the set ci,j

(3) A + (B + C) = the set ai,j + (bi,j + ci,j)

(4) (A + B) + C = the set (ai,j + bi,j) + ci,j

(5) Since ai,j, bi,j, ci,j are all scalar, ai,j + (bi,j + ci,j) = (ai,j + bi,j) + ci,j

QED

A + Zm,n = Zm,n+ A = A

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

Proof:

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

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

(3) A + Zm,n = the set of ai,j + zi,j = the set of ai,j = A

(4) Zm,n + A = the set of zi,j + ai,j = ai,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 ai,j, let B = the set of bi,j

(2) A + B = the set ai,j + bi,j

(3) c(A+B) = the set c(ai,j + bi,j) = ai,jc + bi,jc .

(4) cA + cB = the set of ai,jc + the set of bi,jc = the set of ai,jc + bi,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 ci,j

(2) (a + b)C = the set of (a + b)ci,j = aci,j + bci,j

(3) aC + bC = the set of aci,j+ the set of bci,j = the set of aci,j + bci,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 ci,j

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

(3) a(bC) = a*(the set of bci,j) = the set of abci,j

QED

References
• Gareth Williams, , Wm. C. Brown Publishers, 1996.