# Questions about Cholesky and QR factorizations

**URL:** <https://discourse.julialang.org/t/questions-about-cholesky-and-qr-factorizations/128728>\
**Category:** Numerics\
**Tags:** linearalgebra\
**Created:** [May 5, 2025, 2:48pm UTC](https://discourse.julialang.org/t/questions-about-cholesky-and-qr-factorizations/128728 "2025-05-05T14:48:36Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![WalterMadelim](https://avatars.discourse-cdn.com/v4/letter/w/3e96dc/32.png) [@WalterMadelim](https://discourse.julialang.org/u/WalterMadelim)\
**Post date:** [May 5, 2025, 2:48pm UTC](https://discourse.julialang.org/t/questions-about-cholesky-and-qr-factorizations/128728/1 "2025-05-05T14:48:36Z")

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I’ve learned that.

(But the title of the topic is very general, which is tempting for people to discuss things.)

But I find that I cannot perform `U \ UQ` (see the `🟠` line) due to  
`ERROR: MethodError: no method matching /(::LinearAlgebra.AdjointQ{Float64, LinearAlgebra.QRCompactWYQ{…}}, ::LowerTriangular{Float64, Adjoint{…}})`.  
Is this normal?  
Another question is, why is the `R` factor not of type `UpperTriangular`? (From the perspective of LinearAlgebra.jl, `UpperTriangular` is not considered a sparse matrix? since I notice that `lu` has the same behavior) Would it be beneficial if I do it myself (see the `🟤` line)?

```julia
using LinearAlgebra
N = 4
P = rand(-9:.17:9, N, N); P = P' * P
U = cholesky(P).U # U' * U == P, Moreover P is posdef if no ERROR occurs
A = rand(-9:.17:9, N, N) # columns in A are linearly independent, verified as follows
rank(A) == N || error("A is not full column rank")
UQ, R = qr(U * A)
R = UpperTriangular(R) # 🟤
Q = U \ Matrix(UQ) # 🟠
Q' * P * Q - I |> norm
Q * R .- A |> norm

```

In this example, columns of `A` are represented with the basis being the columns of `Q`, which are “P-conjugate”—a concept in the context of conjugate gradient method—as testified in the last two lines.

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<div class="post-metadata">

**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [May 5, 2025, 3:58pm UTC](https://discourse.julialang.org/t/questions-about-cholesky-and-qr-factorizations/128728/2 "2025-05-05T15:58:14Z")

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(moved to a new topic … an old thread is not the place for any and all discussions about QR factorization)