# Linear Algebra tricks?
**URL:** https://discourse.julialang.org/t/linear-algebra-tricks/38618
**Category:** General Usage
**Created:** [May 2, 2020, 5:15pm UTC](https://discourse.julialang.org/t/linear-algebra-tricks/38618 "2020-05-02T17:15:05Z")
**Posts on this page:** 9
**Page:** 2
### Author:  [@BLI](https://discourse.julialang.org/u/BLI)
#### Post date: [May 3, 2020, 11:02am UTC](https://discourse.julialang.org/t/linear-algebra-tricks/38618/21 "2020-05-03T11:02:49Z")
OK – I tested it:
```julia
julia> M1 = Eye{Int}(5);
julia> M2 = I(5);
julia> M3 = Diagonal(fill(1,5));
julia> sizeof(M1), sizeof(M2), sizeof(M3)
(8, 8, 8)
julia> sizeof(M1.diag), sizeof(M2.diag), sizeof(M3.diag)
(8, 5, 40)
```
… and then I tested it more:
```julia
julia> M1 = rand(-9:9,3,7)
3×7 Array{Int64,2}:
0 -6 1 3 6 -7 9
-4 -8 8 -5 8 -9 9
8 2 -7 -8 0 -5 -5
julia> M2 = [M1 I]
3×10 Array{Int64,2}:
0 -6 1 3 6 -7 9 1 0 0
-4 -8 8 -5 8 -9 9 0 1 0
8 2 -7 -8 0 -5 -5 0 0 1
julia> sizeof(M1)
168
julia> sizeof(M2)
240
julia> M3 = [M1 Eye(3)]
3×10 SparseArrays.SparseMatrixCSC{Float64,Int64} with 22 stored entries:
[2, 1] = -4.0
[3, 1] = 8.0
[1, 2] = -6.0
[2, 2] = -8.0
[3, 2] = 2.0
[1, 3] = 1.0
[2, 3] = 8.0
[3, 3] = -7.0
[1, 4] = 3.0
[2, 4] = -5.0
[3, 4] = -8.0
[1, 5] = 6.0
[2, 5] = 8.0
[1, 6] = -7.0
[2, 6] = -9.0
[3, 6] = -5.0
[1, 7] = 9.0
[2, 7] = 9.0
[3, 7] = -5.0
[1, 8] = 1.0
[2, 9] = 1.0
[3, 10] = 1.0
julia> sizeof(M2), sizeof(M3)
(240, 40)
```
So – the FillArrays package figured out that `M1` has two zeros, so 19 nonzero elements + the three diagonal elements of `Eye(3)`. That is clever.
---
### Author:  [@DNF](https://discourse.julialang.org/u/DNF)
#### Post date: [May 3, 2020, 11:47am UTC](https://discourse.julialang.org/t/linear-algebra-tricks/38618/22 "2020-05-03T11:47:18Z")
@BLI isn’t this what you were looking for?
```julia
julia> Matrix{Int}(I,3,3)
3×3 Array{Int64,2}:
1 0 0
0 1 0
0 0 1
julia> Matrix{Float64}(I,3,3)
3×3 Array{Float64,2}:
1.0 0.0 0.0
0.0 1.0 0.0
0.0 0.0 1.0
julia> Matrix{ComplexF32}(I,3,3)
3×3 Array{Complex{Float32},2}:
1.0+0.0im 0.0+0.0im 0.0+0.0im
0.0+0.0im 1.0+0.0im 0.0+0.0im
0.0+0.0im 0.0+0.0im 1.0+0.0im
# etc. etc.
```
---
### Author:  [@Skoffer](https://discourse.julialang.org/u/Skoffer)
#### Post date: [May 3, 2020, 11:53am UTC](https://discourse.julialang.org/t/linear-algebra-tricks/38618/23 "2020-05-03T11:53:40Z")
You can also use function `Base.summarysize` which compute the amount of memory, in bytes, used by all unique objects reachable from the argument.
```julia
julia> M1 = Eye{Int}(5);
julia> M2 = I(5);
julia> M3 = Diagonal(fill(1,5));
julia> Base.summarysize(M1), Base.summarysize(M2), Base.summarysize(M3)
(8, 53, 88)
```
and also
```julia
julia> M1 = rand(-9:9,3,7);
julia> M2 = [M1 I];
julia> M3 = [M1 Eye(3)];
julia> Base.summarysize(M1), Base.summarysize(M2), Base.summarysize(M3)
(208, 280, 616)
```
---
### Author:  [@BLI](https://discourse.julialang.org/u/BLI)
#### Post date: [May 3, 2020, 3:01pm UTC](https://discourse.julialang.org/t/linear-algebra-tricks/38618/24 "2020-05-03T15:01:07Z")
Well, I want to make a function dependent on what types the user has chosen, so I guess in my function, I’ll parameterize the type.
---
### Author:  [@BLI](https://discourse.julialang.org/u/BLI)
#### Post date: [May 3, 2020, 3:22pm UTC](https://discourse.julialang.org/t/linear-algebra-tricks/38618/25 "2020-05-03T15:22:52Z")
Final question: if the function doesn’t already exist in the Julia eco-system, I’d like to make a function to “swap” permutation vectors. What I mean… suppose I have a vector `cp` (“column permutation”) such that `A[:,cp] = A*P` where `P = I(length(cp))[:,cp]` is a permutation matrix. Then I’d like to compute the permutation vector `rp` so that `A[rp,:] = P*A`. The following function will do the trick:
```julia
function permswap(v::Vector{T}) where T <: Integer
n = length(v)
vsort = sort(v)
if vsort == 1:n
return I(n)[:,v]*vsort
else
error("Argument is not a permutation vector")
end
end
```
Since using this function twice brings be back to the original, i.e., `permswap(permswap(v)) = v`, it can be used both for converting `cp` to `rp` and vice versa.
**Question** – since I’m a beginner when it comes to making functions in Julia: what would be a better name for this function? Would `perm_swap` be better? Or `permutation_swap`? Or `swapperm`? Or something else? (Note: there is already a `randperm` function.)
---
### Author:  [@dlfivefifty](https://discourse.julialang.org/u/dlfivefifty)
#### Post date: [May 3, 2020, 6:04pm UTC](https://discourse.julialang.org/t/linear-algebra-tricks/38618/26 "2020-05-03T18:04:29Z")
It works that way because `Eye(5)` is shorthand for `Diagonal(Ones(5))`, where `Ones(5)` is a lazy vector of all ones. So anything `Diagonal` can do, so can `Eye`, in this case making a sparse array.
Note FillArrays.jl types are immutable.
---
### Author:  [@stevengj](https://discourse.julialang.org/u/stevengj)
#### Post date: [May 3, 2020, 6:30pm UTC](https://discourse.julialang.org/t/linear-algebra-tricks/38618/27 "2020-05-03T18:30:14Z")
> [@BLI](#):
>
> Final question: if the function doesn’t already exist in the Julia eco-system, I’d like to make a function to “swap” permutation vectors.
`invperm(v)` does what you want, I think. (Transposing a permutation matrix is equivalent to inverting it, since permutations are orthogonal/unitary.)
The [implementation of `invperm`](https://github.com/JuliaLang/julia/blob/cf3d8b584a44d48b0dc66db9ab37f871f63a21c7/base/combinatorics.jl#L259-L269) might be worth looking over — it should be quite a bit faster than your `permswap` function. (Your version constructs an `n x n` dense matrix `I(n)`, then constructs a new matrix with re-ordered columns, then performs a matrix-vector multiplication, for O(n²) space and time complexity, whereas `invperm` is O(n).)
---
### Author:  [@BLI](https://discourse.julialang.org/u/BLI)
#### Post date: [May 3, 2020, 6:48pm UTC](https://discourse.julialang.org/t/linear-algebra-tricks/38618/28 "2020-05-03T18:48:00Z")
Indeed! `invperm()` is some 5-6000 times faster than my hobby algorithm. [How does one go about finding such routines?]
---
### Author:  [@Tamas\_Papp](https://discourse.julialang.org/u/Tamas_Papp)
#### Post date: [May 4, 2020, 5:59am UTC](https://discourse.julialang.org/t/linear-algebra-tricks/38618/29 "2020-05-04T05:59:16Z")
Eg
```julia
using LinearAlgebra
apropos("permutation")
```
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