How to concatenate matrices in diagonal form in Julia

Question

I have a matrix A and I want to build a block diagonal matrix M as

A 0 ⋯ 0
0 A ⋯ 0
⋮ ⋮ ⋱ ⋮
0 0 ⋯ A

where the 0s are matrices of the same size as A filled with zeros. Is there a convenient way to do that?

In other words, is there an equivalent to Python's linalg.block_diag (from scipy)?

Answer 1

You can use blockdiag from the SparseArrays.jl standard library. For example:

julia> using SparseArrays

julia> A = sparse(rand(3,2)) # A must be sparse for blockdiag
3×2 SparseMatrixCSC{Float64, Int64} with 6 stored entries:
 0.465226  0.473656
 0.230678  0.391924
 0.928409  0.772551

julia> M = blockdiag(A, A, A)
9×6 SparseMatrixCSC{Float64, Int64} with 18 stored entries:
 0.465226  0.473656   ⋅         ⋅         ⋅         ⋅
 0.230678  0.391924   ⋅         ⋅         ⋅         ⋅
 0.928409  0.772551   ⋅         ⋅         ⋅         ⋅
  ⋅         ⋅        0.465226  0.473656   ⋅         ⋅
  ⋅         ⋅        0.230678  0.391924   ⋅         ⋅
  ⋅         ⋅        0.928409  0.772551   ⋅         ⋅
  ⋅         ⋅         ⋅         ⋅        0.465226  0.473656
  ⋅         ⋅         ⋅         ⋅        0.230678  0.391924
  ⋅         ⋅         ⋅         ⋅        0.928409  0.772551

And if you really need M to be dense, you can just convert it back with

julia> Matrix(M)
9×6 Matrix{Float64}:
 0.465226  0.473656  0.0       0.0       0.0       0.0
 0.230678  0.391924  0.0       0.0       0.0       0.0
 0.928409  0.772551  0.0       0.0       0.0       0.0
 0.0       0.0       0.465226  0.473656  0.0       0.0
 0.0       0.0       0.230678  0.391924  0.0       0.0
 0.0       0.0       0.928409  0.772551  0.0       0.0
 0.0       0.0       0.0       0.0       0.465226  0.473656
 0.0       0.0       0.0       0.0       0.230678  0.391924
 0.0       0.0       0.0       0.0       0.928409  0.772551

Answer 2

You can do this with cat, giving it two dimensions:

julia> A = [1 2; 3 4]
2×2 Matrix{Int64}:
 1  2
 3  4

julia> cat(A, A', reverse(A); dims=(1,2))
6×6 Matrix{Int64}:
 1  2  0  0  0  0
 3  4  0  0  0  0
 0  0  1  3  0  0
 0  0  2  4  0  0
 0  0  0  0  4  3
 0  0  0  0  2  1