Definitition 1: Matrix
A matrix is an ordering of elements into rows and columns. A matrix M that is an r x c matrix consists of r rows and c columns.
For example, a 2 x 3 matrix contains 6 elements ordered into 2 rows and 3 columns.
I will refer to a matrix as a capital letter and the elements that make up a matrix as a lowercase letter followed by a subscript that specifies row and column.
For example, a 2 x 3 matrix M consists of the following elements: m1,1, m1,2, m1,3, m2,1, m2,2, m2,3 that would look like this:
So that we have, M = [ mi,j ]2 x 3
Definition 2: Matrix Equality
If A is an m x n matrix and B is a p x q matrix, then A = B if and only if m=p and n=q and ai,j = bi,j for all i in (1 .. m) and for all j in (1 .. n).
In other words, two matrices are equal if they consist of the same number of rows and columns and all their elements are equal.
Definition 3: Matrix Addition
If A, B are two matrices of the same dimensions m x n, then the sum C = A + B is given by:
ci,j = ai,j + bi,j for all i in (1 .. m) and all j in (1 .. n)
where C is also an m x n matrix.
Matrices can be only be added together when they consist of the same number of rows and columns.
Definition 4: Scalar
A scalar is any element of a matrix which is a number.
In this blog, I have made the assumption that all the elements of a matrix are scalar.
I will represent a scalar value as a lowercase letter that does not have a subscript.
Definition 5: Scalar Multiplication
If a is a scalar and B is an m x n matrix, then C = aB is also an m x n matrix such that:
ci,j = a*bi,j for all i in (1..m) and for all j in (1 .. n).
In my next blog, I will review matrix multiplication.
- Hans Schneider, George Philip Barker, Matrices and Linear Algebra, 1989.