# Matrix division \`\\\` vs compute by hand, why such big difference?

**URL:** <https://discourse.julialang.org/t/matrix-division-vs-compute-by-hand-why-such-big-difference/98288>\
**Category:** General Usage\
**Tags:** linearalgebra\
**Created:** [May 4, 2023, 1:32am UTC](https://discourse.julialang.org/t/matrix-division-vs-compute-by-hand-why-such-big-difference/98288 "2023-05-04T01:32:47Z")\
**Posts on this page:** 1\
**Page:** 1

<div class="post-metadata">

**Author:** ![maxchendt](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/maxchendt/32/42976_2.png) [@maxchendt](https://discourse.julialang.org/u/maxchendt)\
**Post date:** [May 4, 2023, 1:32am UTC](https://discourse.julialang.org/t/matrix-division-vs-compute-by-hand-why-such-big-difference/98288/1 "2023-05-04T01:32:47Z")

</div>

In the following example, it seems that the Matrix division `\` does not provide an intended computation.

Let `x= A'\b`, then if `norm(A'*x-b, Inf)` is zero such that `A'x-b == 0`, then `b` is a linear combination of rows from `A` (`b` lies in the row space of `A`).

I am using Matrix division `\` to test if `b` lies in the row space of `A` or not, and find that the `\` gives a result different from the result by the _reduced row echelon form_. Hence, I compute it by hand, the `norm(A'*x-b, Inf)` is 4.755666482359269e-12, but the result from Matrix division is 0.00010237, far from zero (tolerance)

Download [matH.jls](https://github.com/maxchendt/FuturesHedge/raw/master/data/matH.jls) first.

```julia
using LinearAlgebra, Serialization

function loadData(fn)
    #load data
    local t
    open(fn, "r") do io
        t = deserialize(io)
    end
    return t
end

function main()
    H, v = loadData("./matH.jls")

    G = qr(H', ColumnNorm())
    z = G.P*inv(G.R)*Matrix(G.Q)'*v
    display(norm(H'*z-v, Inf)) #4.755666482359269e-12

    z1 = G\v
    display(norm(H'*z1-v, Inf)) #0.00010237343843657264

end

main()

nothing

```

From the [\(A,B)](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#Base.:%5C-Tuple%7BAbstractMatrix,%20AbstractVecOrMat%7D) , the docstring of Julia: " For rectangular `A` the result is the minimum-norm least squares solution computed by a pivoted QR factorization of `A` and a rank estimate of `A` based on the R factor."

Is there any setting I can choose when using Matrix division `\` in Julia?

Edit: more discussions on [`x=A\b` by `qr(A, ColumnNorm()) \ B`, Matrix division does not yield `Ax=b` · Issue #49625 · JuliaLang/julia · GitHub](https://github.com/JuliaLang/julia/issues/49625)