# Leading principal submatrix of Cholesky factor
**URL:** https://discourse.julialang.org/t/leading-principal-submatrix-of-cholesky-factor/72456
**Category:** Numerics
**Tags:** question, linearalgebra
**Created:** [December 2, 2021, 2:56pm UTC](https://discourse.julialang.org/t/leading-principal-submatrix-of-cholesky-factor/72456 "2021-12-02T14:56:25Z")
**Posts on this page:** 4
**Page:** 1
### Author:  [@DanielVandH](https://discourse.julialang.org/u/DanielVandH)
#### Post date: [December 2, 2021, 2:56pm UTC](https://discourse.julialang.org/t/leading-principal-submatrix-of-cholesky-factor/72456/1 "2021-12-02T14:56:25Z")
I have an algorithm that uses only the first k rows and k columns of a Cholesky factor L (i.e., the kth leading principal submatrix L = L[1:k, 1:k]). Currently I just extract it after the computation of the whole Cholesky factor, but is there any way to have `cholesky` from `LinearAlgebra` compute only these elements given some positive definite matrix A?
Moreover, is there a way within Julia to efficiently compute the dot product of the kth row of the Cholesky factor with another vector of length k, or at least compute just the kth row efficiently? Or is the optimal way just taking this principal submatrix above and extracting L[k, 1:k]?
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### Author:  [@stevengj](https://discourse.julialang.org/u/stevengj)
#### Post date: [December 2, 2021, 6:15pm UTC](https://discourse.julialang.org/t/leading-principal-submatrix-of-cholesky-factor/72456/2 "2021-12-02T18:15:51Z")
> [@DanielVandH](#):
>
> I have an algorithm that uses only the first k rows and k columns of a Cholesky factor L (i.e., the kth leading principal submatrix L = L[1:k, 1:k]). Currently I just extract it after the computation of the whole Cholesky factor, but is there any way to have `cholesky` from `LinearAlgebra` compute only these elements given some positive definite matrix A?
Yup, just compute the Cholesky factor of the submatrix. For example, here is the 5x5 leading principal submatrix of the Cholesky factor of the 100x100 SPD matrix `A = rand(100,100); A = A'A`, computed the slow way and the fast way:
```julia
julia> cholesky(A).L[1:5,1:5] # slow way: form whole Cholesky factor, then take submatrix
5×5 Matrix{Float64}:
5.81234 0.0 0.0 0.0 0.0
4.50388 3.68507 0.0 0.0 0.0
4.1697 1.13275 3.27478 0.0 0.0
3.77458 1.3239 1.19011 3.14419 0.0
4.65957 1.30715 1.30752 0.63263 2.95488
julia> cholesky(@view A[1:5,1:5]).L # fast way: form Cholesky factor from submatrix of A
5×5 LowerTriangular{Float64, Matrix{Float64}}:
5.81234 ⋅ ⋅ ⋅ ⋅
4.50388 3.68507 ⋅ ⋅ ⋅
4.1697 1.13275 3.27478 ⋅ ⋅
3.77458 1.3239 1.19011 3.14419 ⋅
4.65957 1.30715 1.30752 0.63263 2.95488
```
This is a consequence of how Cholesky factorization is defined: a leading principal submatrix of the Cholesky factor of A is always equivalent to a Cholesky factor of the corresponding leading principal submatrix of A.
> [@DanielVandH](#):
>
> Moreover, is there a way within Julia to efficiently compute the dot product of the kth row of the Cholesky factor with another vector of length k, or at least compute just the kth row efficiently?
From above, just do `@views dot(cholesky(A[1:k,1:k]).L[k,1:k], vec)`.
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### Author:  [@DanielVandH](https://discourse.julialang.org/u/DanielVandH)
#### Post date: [December 2, 2021, 10:52pm UTC](https://discourse.julialang.org/t/leading-principal-submatrix-of-cholesky-factor/72456/3 "2021-12-02T22:52:47Z")
Very simple! Thank you. What is the idea of using `@views` here? The documentation isn’t too clear, does it help with memory allocation?
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### Author:  [@stevengj](https://discourse.julialang.org/u/stevengj)
#### Post date: [December 3, 2021, 1:03am UTC](https://discourse.julialang.org/t/leading-principal-submatrix-of-cholesky-factor/72456/4 "2021-12-03T01:03:09Z")
> [@DanielVandH](#):
>
> What is the idea of using `@views` here? The documentation isn’t too clear, does it help with memory allocation?
Yes, it prevents a copy being made for a slice like `A[1:k,1:k]`. See:
- [Views (Subarrays and other view types)](https://docs.julialang.org/en/v1/base/arrays/#Views-(SubArrays-and-other-view-types))
- [Consider using views for slices](https://docs.julialang.org/en/v1/manual/performance-tips/#man-performance-views)