Functions for 2D Array Operations in Julia
Let’s say $A = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 3 & 0 \\ 2 & 3 & 4\end{pmatrix}$.
Transpose Matrix
julia> A =[1 2 1;
0 3 0;
2 3 4]
3×3 Array{Int64,2}:
1 2 1
0 3 0
2 3 4
julia> transpose(A)
3×3 LinearAlgebra.Transpose{Int64,Array{Int64,2}}:
1 0 2
2 3 3
1 0 4
julia> A'
3×3 LinearAlgebra.Adjoint{Int64,Array{Int64,2}}:
1 0 2
2 3 3
1 0 4
When the elements of a matrix are real numbers, transpose() and ' return the same matrix, but the data type is subtly different. This is because ' is not exactly transpose but conjugate transpose. Therefore, for real matrices, it effectively returns the same matrix, but for complex matrices, it returns a completely different result.
julia> A_complex=[1+im 2 1+im;
0 3 0+im;
2 3+im 4]
3×3 Array{Complex{Int64},2}:
1+1im 2+0im 1+1im
0+0im 3+0im 0+1im
2+0im 3+1im 4+0im
julia> transpose(A_complex)
3×3 LinearAlgebra.Transpose{Complex{Int64},Array{Complex{Int64},2}}:
1+1im 0+0im 2+0im
2+0im 3+0im 3+1im
1+1im 0+1im 4+0im
julia> A_complex'
3×3 LinearAlgebra.Adjoint{Complex{Int64},Array{Complex{Int64},2}}:
1-1im 0+0im 2+0im
2+0im 3+0im 3-1im
1-1im 0-1im 4+0im
Power
julia> A =[1 2 1;
0 3 0;
2 3 4]
3×3 Array{Int64,2}:
1 2 1
0 3 0
2 3 4
julia> A^2
3×3 Array{Int64,2}:
3 11 5
0 9 0
10 25 18
julia> A*A
3×3 Array{Int64,2}:
3 11 5
0 9 0
10 25 18
julia> A^3
3×3 Array{Int64,2}:
13 54 23
0 27 0
46 149 82
julia> A*A*A
3×3 Array{Int64,2}:
13 54 23
0 27 0
46 149 82
A^2 and A*A return exactly the same result. Likewise, A^3 and A*A*A are the same.
Element-wise Multiplication, Element-wise Division
julia> A =[1 2 1;
0 3 0;
2 3 4]
3×3 Array{Int64,2}:
1 2 1
0 3 0
2 3 4
julia> A.*A
3×3 Array{Int64,2}:
1 4 1
0 9 0
4 9 16
julia> A./A
3×3 Array{Float64,2}:
1.0 1.0 1.0
NaN 1.0 NaN
1.0 1.0 1.0
Returns the result of multiplying or dividing each element.
Horizontal Flip, Vertical Flip
julia> A =[1 2 1;
0 3 0;
2 3 4]
3×3 Array{Int64,2}:
1 2 1
0 3 0
2 3 4
julia> reverse(A,dims=1)
3×3 Array{Int64,2}:
2 3 4
0 3 0
1 2 1
julia> reverse(A,dims=2)
3×3 Array{Int64,2}:
1 2 1
0 3 0
4 3 2
reverse(A,dims=1) returns the vertically flipped matrix of $A$ and is equivalent to flipud(A) in MATLAB. reverse(A,dims=2) returns the horizontally flipped matrix of $A$ and is equivalent to fliplr(A) in MATLAB.
Inverse Matrix
julia> A =[1 2 1;
0 3 0;
2 3 4]
3×3 Array{Int64,2}:
1 2 1
0 3 0
2 3 4
julia> inv(A)
3×3 Array{Float64,2}:
2.0 -0.833333 -0.5
0.0 0.333333 0.0
-1.0 0.166667 0.5
Returns the inverse matrix of $A$. If the inverse matrix cannot be found, it throws an error.
Environment
- OS: Windows10
- Version: 1.5.3 (2020-11-09)