# Does Julia recognize block, banded, and block-banded matrices?
**URL:** https://discourse.julialang.org/t/does-julia-recognize-block-banded-and-block-banded-matrices/120821
**Category:** Numerics
**Tags:** question, linearalgebra
**Created:** [October 2, 2024, 6:28pm UTC](https://discourse.julialang.org/t/does-julia-recognize-block-banded-and-block-banded-matrices/120821 "2024-10-02T18:28:21Z")
**Posts on this page:** 7
**Page:** 1
### Author:  [@rsenne](https://discourse.julialang.org/u/rsenne)
#### Post date: [October 2, 2024, 6:28pm UTC](https://discourse.julialang.org/t/does-julia-recognize-block-banded-and-block-banded-matrices/120821/1 "2024-10-02T18:28:21Z")
For a simple example of why I need this, let’s say I want to compute the entropy of a multivariate Gaussian. One can do this as follows:
\text{Entropy} \propto -\frac{1}{2}\*\log|-H|
where -H is the negative Hessian at the maximum of the Gaussian distribution. I.e., we need to calculate the log-determinant of the negative Hessian. One can calculate this in a stable way using the Cholesky decomposition, and in theory this can be done in O(T) time but only if you exploit the structure of the matrix (in my case this would be a banded matrix).
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### Author:  [@Oscar\_Smith](https://discourse.julialang.org/u/Oscar_Smith)
#### Post date: [October 2, 2024, 6:32pm UTC](https://discourse.julialang.org/t/does-julia-recognize-block-banded-and-block-banded-matrices/120821/2 "2024-10-02T18:32:13Z")
See BandedMatrices.jl
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### Author:  [@DanielVandH](https://discourse.julialang.org/u/DanielVandH)
#### Post date: [October 2, 2024, 6:35pm UTC](https://discourse.julialang.org/t/does-julia-recognize-block-banded-and-block-banded-matrices/120821/3 "2024-10-02T18:35:30Z")
BandedMatrices.jl, BlockArrays.jl, BlockBandedMatrices.jl. Also LazyBandedMatrices.jl is available.
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### Author:  [@rsenne](https://discourse.julialang.org/u/rsenne)
#### Post date: [October 2, 2024, 6:42pm UTC](https://discourse.julialang.org/t/does-julia-recognize-block-banded-and-block-banded-matrices/120821/4 "2024-10-02T18:42:05Z")
Do you know if these packages are compatible with other LinearAlgebra routines? It seems Banded.jl has some methods that override things like the `\` operator, but for something like:
`cholesky(my_banded_matrix`)
or some other general method. I guess in the same way using `Hermitian(some_matrix)` from LinearAlgebra.jl provides speed-ups if the specific structure can be exploited?
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### Author:  [@DanielVandH](https://discourse.julialang.org/u/DanielVandH)
#### Post date: [October 2, 2024, 6:43pm UTC](https://discourse.julialang.org/t/does-julia-recognize-block-banded-and-block-banded-matrices/120821/5 "2024-10-02T18:43:02Z")
Using the interfaces from ArrayLayouts.jl, yes they should e.g.
```julia
julia> using BandedMatrices, LinearAlgebra, ArrayLayouts
julia> A = Symmetric(brand(10,10,2,2) + 10I);
julia> cholesky(A) # using a banded chol
Cholesky{Float64, BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}}
U factor:
10×10 UpperTriangular{Float64, BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}}:
3.17267 0.105034 0.144806 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ 3.19379 0.0523953 0.251397 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ 3.26506 0.151593 0.0750854 ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ 3.26117 0.0102327 0.0144174 ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ 3.16206 0.181666 0.107728 ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ 3.21687 0.0727688 0.117991 ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 3.23698 0.0547971 0.191308 ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 3.19253 0.0834566 0.278381
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 3.16049 0.229518
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 3.21334
julia> ArrayLayouts.cholesky_layout(MemoryLayout(A), axes(A), A)
Cholesky{Float64, BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}}
U factor:
10×10 UpperTriangular{Float64, BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}}:
3.17267 0.105034 0.144806 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ 3.19379 0.0523953 0.251397 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ 3.26506 0.151593 0.0750854 ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ 3.26117 0.0102327 0.0144174 ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ 3.16206 0.181666 0.107728 ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ 3.21687 0.0727688 0.117991 ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 3.23698 0.0547971 0.191308 ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 3.19253 0.0834566 0.278381
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 3.16049 0.229518
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 3.21334
```
```julia
julia> @which ArrayLayouts.cholesky_layout(MemoryLayout(A), axes(A), A)
cholesky_layout(::Union{HermitianLayout{<:AbstractBandedLayout}, SymmetricLayout{<:AbstractBandedLayout}}, ax, A; ...)
@ BandedMatrices C:\Users\danjv\.julia\packages\BandedMatrices\WmDeb\src\symbanded\BandedCholesky.jl:96
```
Same goes for the other packages.
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### Author:  [@rsenne](https://discourse.julialang.org/u/rsenne)
#### Post date: [October 2, 2024, 6:48pm UTC](https://discourse.julialang.org/t/does-julia-recognize-block-banded-and-block-banded-matrices/120821/6 "2024-10-02T18:48:32Z")
Oh great thanks! Wasn’t aware of the existence of ArrayLayouts.jl. Will give it a go.
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### Author:  [@DanielVandH](https://discourse.julialang.org/u/DanielVandH)
#### Post date: [October 2, 2024, 6:49pm UTC](https://discourse.julialang.org/t/does-julia-recognize-block-banded-and-block-banded-matrices/120821/7 "2024-10-02T18:49:24Z")
(Note that you don’t need to interact with ArrayLayouts.jl directly, just showing that the methods you don’t see overloaded are because of this interface, so `cholesky(A)` will just work with the most efficient method.)