Simple matrix operations

We will be meeting matrices and matrix operations throughout this book, but let us look now at the simplest of operations.

Let’s take A and B, as defined in the following code snippet:

julia> A = [1 2 3; 4 5 6];
julia> B = [1 5; 4 3; 2 6];

The normal matrix rules apply, which is a feature of multiple dispatch; we will cover this in Chapter 4.

The transpose of B can be computed as follows:

julia> C = transpose(B)
2×3 transpose(::Matrix{Int64}) with eltype Int64
1 4 2
5 3 6

This can also be written more compactly as C = B’:

julia> A + C
2x3 Matrix{Int64}:
2 6 5
9 8 12
julia> A*B
2x2 Matrix{Int64}:
15 29
36 71

Matrix division makes more sense with square matrices, but it is possible to define the operations for non-square matrices too. Note here that the / and \ operations produce results of different sizes:

julia> A / C
2x2 Matrix{Float64}
0.332273 0.27663
0.732909 0.710652
julia> A \ C
3x3 Matrix{Float64}:
 1.27778  -2.44444  0.777778
 0.444444 -0.111111 0.444444
-0.388889  2.22222  0.111111

The type of the array was previously defined as Array{Int64,2} rather than the now more compact form of Matrix{Int64}, and ditto Array{Float64,2} has been replaced with Matrix{Float64}.

We will discuss matrix decomposition in more detail later when looking at linear algebra.

Although A * C is not allowed because the number of columns of A is not equal to the number of rows of C, the following broadcasts are all valid:

julia> A .* C 2x3 Matrix{Int64}: 1  8  6 20 15 36
julia> A ./ C 2x3 Matrix{Float64}: 1.0 0.5     1.5 0.8 1.66667 1.0
julia> A .== C 2x3 BitMatrix 1  0  0 0  0  1

So far, we have only been looking at manipulating variables representing arithmetic values. Julia has a variety of string types, which we will look at next.