feat(matrices): add minij

Author: AnzhiZhang

src/matrices/index.jl

@@ -1,12 +1,15 @@
 export
     list_groups,
-    list_matrices
+    list_matrices,
+    Minij
 
 # include all matrices
+include("minij.jl")
 
 # matrix groups
 const MATRIX_GROUPS = Dict{Group,Set{Type{<:AbstractMatrix}}}()
 MATRIX_GROUPS[Group(:builtin)] = Set([
+    Minij,
 ])
 MATRIX_GROUPS[Group(:user)] = Set([])
 

src/matrices/minij.jl

@@ -0,0 +1,56 @@
+"""
+MIN[I,J] Matrix
+===============
+A matrix with `(i,j)` entry `min(i,j)`. It is a symmetric positive
+     definite matrix. The eigenvalues and eigenvectors are known
+     explicitly. Its inverse is tridiagonal.
+
+*Input options:*
+
++ [type,] dim: the dimension of the matrix.
+
+*Groups:* ["inverse", "symmetric", "posdef", "eigen"]
+
+*References:*
+
+**J. Fortiana and C. M. Cuadras**, A family of matrices,
+            the discretized Brownian bridge, and distance-based regression,
+            Linear Algebra Appl., 264 (1997), 173-188.  (For the eigensystem of A.)
+"""
+struct Minij{T<:Integer} <: AbstractMatrix{T}
+    n::Int
+
+    function Minij(::Type{T}, n::Int) where {T<:Integer}
+        n > 0 || throw(ArgumentError("$n ≤ 0"))
+        return new{T}(n)
+    end
+end
+
+# constructors
+Minij(n::Int) = Minij(Int, n)
+
+# metadata
+@properties Minij [:symmetric, :inverse, :posdef, :eigen]
+
+# properties
+size(s::Minij) = (s.n, s.n)
+LinearAlgebra.isdiag(::Minij) = false
+LinearAlgebra.ishermitian(::Minij) = true
+LinearAlgebra.isposdef(::Minij) = true
+LinearAlgebra.issymmetric(::Minij) = true
+LinearAlgebra.adjoint(A::Minij) = A
+LinearAlgebra.transpose(A::Minij) = A
+
+# functions
+@inline Base.Base.@propagate_inbounds function getindex(A::Minij{T}, i::Integer, j::Integer) where {T}
+    @boundscheck checkbounds(A, i, j)
+    return T(min(i, j))
+end
+
+function LinearAlgebra.inv(A::Minij{T}) where {T}
+    if A.n == 1
+        return ones(T, 1, 1)
+    else
+        return SymTridiagonal(2 * ones(T, A.n), -ones(T, A.n - 1))
+    end
+end