# How can I use \\= with factorizations?

**URL:** <https://discourse.julialang.org/t/how-can-i-use-with-factorizations/59126>\
**Category:** General Usage\
**Created:** [April 12, 2021, 5:47pm UTC](https://discourse.julialang.org/t/how-can-i-use-with-factorizations/59126 "2021-04-12T17:47:56Z")\
**Posts on this page:** 8\
**Page:** 1

<div class="post-metadata">

**Author:** ![giancarloantonucci](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/giancarloantonucci/32/202551_2.png) [@giancarloantonucci](https://discourse.julialang.org/u/giancarloantonucci)\
**Post date:** [April 12, 2021, 5:47pm UTC](https://discourse.julialang.org/t/how-can-i-use-with-factorizations/59126/1 "2021-04-12T17:47:56Z")

</div>

As per title.

For example, this works:

```julia
using LinearAlgebra
A = rand(3, 3)
b = rand(3)
@show b
b \= A
@show b

```

```julia
b = [0.09019616214492654, 0.2564457656075061, 0.7120782534101571]
b = [0.5468125936523385 1.4817879951917017 0.5732261599468792]

```

but this doesn’t:

```julia
using LinearAlgebra
A = factorize(rand(3, 3))
b = rand(3)
@show b
b \= A
@show b

```

```julia
b = [0.54431634700671, 0.6989662716060527, 0.244305881937394]
StackOverflowError:

Stacktrace:
     [1] \(x::Vector{Float64}, y::LU{Float64, Matrix{Float64}})
       @ Base ./operators.jl:598
     [2] /(A::Adjoint{Float64, LU{Float64, Matrix{Float64}}}, B::Adjoint{Float64, Vector{Float64}})
       @ LinearAlgebra /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.6/LinearAlgebra/src/generic.jl:1144
--- the last 2 lines are repeated 25974 more times ---
 [51951] \(x::Vector{Float64}, y::LU{Float64, Matrix{Float64}})
       @ Base ./operators.jl:598
 [51952] top-level scope
       @ In[69]:5
 [51953] eval
       @ ./boot.jl:360 [inlined]

```

What’s the easiest way to make the second one work too? Should I add a new method to `ldiv!`? Would that be enough?

---

<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:** [April 12, 2021, 5:52pm UTC](https://discourse.julialang.org/t/how-can-i-use-with-factorizations/59126/2 "2021-04-12T17:52:25Z")

</div>

> [@giancarloantonucci](#):
>
> For example, this works:

It gives an answer, but I’m not sure it does what you think it does.

`b \= A` is equivalent to `b = b \ A`. This means two things:

1. It is essentially equivalent to `pinv(b) * A` — that is, you are dividing `A` by `b`, not `b` by `A`. Did you want `b' / A` instead, i.e. b^T A^{-1}?

2. It is not in-place. It first computes `b \ A`, allocating a new row vector for the result, and then assigns the result to `b` (discarding the original column vector).

---

<div class="post-metadata">

**Author:** ![giancarloantonucci](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/giancarloantonucci/32/202551_2.png) [@giancarloantonucci](https://discourse.julialang.org/u/giancarloantonucci)\
**Post date:** [April 12, 2021, 5:58pm UTC](https://discourse.julialang.org/t/how-can-i-use-with-factorizations/59126/3 "2021-04-12T17:58:22Z")

</div>

Are you sure about (1.)? `pinv(b) * A` gives me a `DimensionMismatch("A has dimensions (3,1) but B has dimensions (3,3)")`. I assumed that was equivalent to `b = A \ b`, which dimensionally makes sense.

---

<div class="post-metadata">

**Author:** ![gustaphe](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/gustaphe/32/18174_2.png) [@gustaphe](https://discourse.julialang.org/u/gustaphe)\
**Post date:** [April 12, 2021, 6:05pm UTC](https://discourse.julialang.org/t/how-can-i-use-with-factorizations/59126/4 "2021-04-12T18:05:58Z")

</div>

The docstring says you’re wrong, so if it does something else that’s probably a bug.

---

<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:** [April 12, 2021, 6:07pm UTC](https://discourse.julialang.org/t/how-can-i-use-with-factorizations/59126/5 "2021-04-12T18:07:46Z")

</div>

> [@giancarloantonucci](#):
>
> `pinv(b) * A` gives me a `DimensionMismatch("A has dimensions (3,1) but B has dimensions (3,3)")` . I assumed that was equivalent to `b = A \ b` , which dimensionally makes sense.

Quite sure:

```julia
julia> A = rand(3, 3); b = rand(3);

julia> pinv(b)*A
1×3 transpose(::Vector{Float64}) with eltype Float64:
 0.998627 1.23721 1.55637

julia> b \ A
1×3 transpose(::Vector{Float64}) with eltype Float64:
 0.998627 1.23721 1.55637

```

I think you were running this _after_ you ran `b \= A`, which replaced `b` (a column vector) with the row vector `b \ A`. And using a row vector, e.g. `pinv(b')*A`, indeed gives a dimension-mismatch error, because this doesn’t work with the [definition of the pseudo-inverse](https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse).

---

<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:** [April 12, 2021, 6:09pm UTC](https://discourse.julialang.org/t/how-can-i-use-with-factorizations/59126/6 "2021-04-12T18:09:26Z")

</div>

> [@giancarloantonucci](#):
>
> I assumed that was equivalent to `b = A \ b` ,

This is never true in Julia. `x OP= y` is _always_ equivalent to `x = x OP y`.

---

<div class="post-metadata">

**Author:** ![giancarloantonucci](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/giancarloantonucci/32/202551_2.png) [@giancarloantonucci](https://discourse.julialang.org/u/giancarloantonucci)\
**Post date:** [April 12, 2021, 6:10pm UTC](https://discourse.julialang.org/t/how-can-i-use-with-factorizations/59126/7 "2021-04-12T18:10:05Z")

</div>

Ah! I see. Then the question comes down to: Is there a way to add a new definition for `\=`?

---

<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:** [April 12, 2021, 6:11pm UTC](https://discourse.julialang.org/t/how-can-i-use-with-factorizations/59126/8 "2021-04-12T18:11:07Z")

</div>

> [@giancarloantonucci](#):
>
> Ah! I see. Then the question comes down to: Is there a way to add a new definition for `\=` ?

No. `x \= y` is just [syntactic sugar](https://en.wikipedia.org/wiki/Syntactic_sugar) for `x = x \ y`. There is no way to redefine `\=` by itself.