# Pseudo-inverse and the backslash operator
**URL:** https://discourse.julialang.org/t/pseudo-inverse-and-the-backslash-operator/27882
**Category:** Optimization (Mathematical)
**Created:** [August 23, 2019, 9:43am UTC](https://discourse.julialang.org/t/pseudo-inverse-and-the-backslash-operator/27882 "2019-08-23T09:43:28Z")
**Posts on this page:** 4
**Page:** 1
### Author:  [@gmouts](https://discourse.julialang.org/u/gmouts)
#### Post date: [August 23, 2019, 9:43am UTC](https://discourse.julialang.org/t/pseudo-inverse-and-the-backslash-operator/27882/1 "2019-08-23T09:43:28Z")
So, I have a sparse matrix `A` and a vector `b` and I want to compute `pinv(A)*b`.
After reading this [post](https://discourse.julialang.org/t/moore-penrose-generalized-inverse-of-sparse-matrix/17414/2) I tried the following:
```julia
using LinearAlgebra
julia> A = [1 1 1; 2 2 2; 1 0 1]
3×3 Array{Int64,2}:
1 1 1
2 2 2
1 0 1
julia> b = [1,0,0]
3-element Array{Int64,1}:
1
0
0
julia> pinv(A)
3×3 Array{Float64,2}:
-2.22045e-16 2.77556e-17 0.5
0.2 0.4 -1.0
-1.52656e-16 1.66533e-16 0.5
julia> pinv(A)*b
3-element Array{Float64,1}:
-2.220446049250313e-16
0.20000000000000048
-1.5265566588595902e-16
```
Everything is fine up to this point, but when I try the backslash operator i get:
```julia
julia> qr(A)\b
3-element Array{Float64,1}:
-1.8385868488076965e15
-1.8242825833464351e-16
1.838586848807696e15
```
I’m not sure what happened here. I don’t understand what the result is supposed to be, but clearly it is not `pinv(A)*b`. Am I using this wrongly or is it a bug?
---
### Author:  [@saschatimme](https://discourse.julialang.org/u/saschatimme)
#### Post date: [August 23, 2019, 9:52am UTC](https://discourse.julialang.org/t/pseudo-inverse-and-the-backslash-operator/27882/2 "2019-08-23T09:52:35Z")
Using a pivoted QR factorization will give you the result you are looking for:
```julia
julia> qr(A, Val(true)) \ b
3-element Array{Float64,1}:
-2.379544922143407e-16
0.20000000000000057
-2.1619865292617238e-16
```
The standard QR algorithm only gives the same result as a multiplication with the pseudo-inverse if the matrix has full column rank.
---
### Author:  [@gmouts](https://discourse.julialang.org/u/gmouts)
#### Post date: [August 23, 2019, 10:01am UTC](https://discourse.julialang.org/t/pseudo-inverse-and-the-backslash-operator/27882/3 "2019-08-23T10:01:52Z")
That works, thanks! However, I’m a bit perplexed by this. What does `qr(A)\b` actually do? What is the result?
---
### Author:  [@saschatimme](https://discourse.julialang.org/u/saschatimme)
#### Post date: [August 23, 2019, 10:31am UTC](https://discourse.julialang.org/t/pseudo-inverse-and-the-backslash-operator/27882/4 "2019-08-23T10:31:44Z")
I think it solve the linear system Rx=Q^Tb but assumes that R has full rank. In the algorithm for the pivoted QR is an explicit check on the rank of R (which is also easier to do there) and then computes a least square solution for the system.