9.2. Special Matrices

using LinearAlgebra

The 2D arrays seen so far have been general, that is, that can be used to represent arbitrary matrices. Many applications lead to matrices with special structures or properties, and Julia defines a number of specialized matrix representations for this. The main reason for using these is to obtain better performance (which can make a very big difference), but they are also useful to e.g. ensure correctness of codes by enforcing known properties.

Type	Description
Symmetric	Symmetric matrix
Hermitian	Hermitian matrix
UpperTriangular	Upper triangular matrix
UnitUpperTriangular	Upper triangular matrix with unit diagonal
LowerTriangular	Lower triangular matrix
UnitLowerTriangular	Lower triangular matrix with unit diagonal
Tridiagonal	Tridiagonal matrix
SymTridiagonal	Symmetric tridiagonal matrix
Bidiagonal	Upper/lower bidiagonal matrix
Diagonal	Diagonal matrix
UniformScaling	Uniform scaling operator

For example, if you know that your matrix is both symmetric and tridiagonal, you can use the SymTridiagonal type. The example below shows how to generate a famous matrix which is very common in applications:

T = SymTridiagonal(2ones(5), -ones(4))

5×5 SymTridiagonal{Float64, Vector{Float64}}:
  2.0  -1.0    ⋅     ⋅     ⋅ 
 -1.0   2.0  -1.0    ⋅     ⋅ 
   ⋅   -1.0   2.0  -1.0    ⋅ 
   ⋅     ⋅   -1.0   2.0  -1.0
   ⋅     ⋅     ⋅   -1.0   2.0

The matrix operations defined above will work just as before on these specialized types, but likely be much more efficient. For example:

T * randn(5)     # Matrix-vector multiplication
T^3              # Matrix cube

5×5 Matrix{Float64}:
  14.0  -14.0    6.0   -1.0    0.0
 -14.0   20.0  -15.0    6.0   -1.0
   6.0  -15.0   20.0  -15.0    6.0
  -1.0    6.0  -15.0   20.0  -14.0
   0.0   -1.0    6.0  -14.0   14.0

9.2.1. The identity matrix

The identity matrix \(I\) is so commonly used, that it has a special syntax which supports some additional performance improvements. In its simplest form, you can simply use the I operator and it will behave as expected:

T + 2I     # OK, since T is 3-by-3

5×5 SymTridiagonal{Float64, Vector{Float64}}:
  4.0  -1.0    ⋅     ⋅     ⋅ 
 -1.0   4.0  -1.0    ⋅     ⋅ 
   ⋅   -1.0   4.0  -1.0    ⋅ 
   ⋅     ⋅   -1.0   4.0  -1.0
   ⋅     ⋅     ⋅   -1.0   4.0

If you want to actually create an identity matrix, you have to specify the element type and the dimensions:

I4 = Matrix{Float64}(I, 4, 4)

4×4 Matrix{Float64}:
 1.0  0.0  0.0  0.0
 0.0  1.0  0.0  0.0
 0.0  0.0  1.0  0.0
 0.0  0.0  0.0  1.0

but this is often not necessary since the I operator can be used in expressions, as shown above.