# Square root of a matrix that has no square root
**URL:** https://discourse.julialang.org/t/square-root-of-a-matrix-that-has-no-square-root/73081
**Category:** General Usage
**Tags:** matrices
**Created:** [December 14, 2021, 3:16pm UTC](https://discourse.julialang.org/t/square-root-of-a-matrix-that-has-no-square-root/73081 "2021-12-14T15:16:49Z")
**Posts on this page:** 4
**Page:** 1
### Author:  [@sijo](https://discourse.julialang.org/u/sijo)
#### Post date: [December 14, 2021, 3:16pm UTC](https://discourse.julialang.org/t/square-root-of-a-matrix-that-has-no-square-root/73081/1 "2021-12-14T15:16:49Z")
Is this expected?
```julia
julia> sqrt([0 1; 0 0])
2×2 Matrix{Float64}:
0.0 0.0
0.0 0.0
```
I thought it would give an error.
---
### Author:  [@Oscar\_Smith](https://discourse.julialang.org/u/Oscar_Smith)
#### Post date: [December 14, 2021, 3:35pm UTC](https://discourse.julialang.org/t/square-root-of-a-matrix-that-has-no-square-root/73081/2 "2021-12-14T15:35:27Z")
Throwing an error here would be hard. The `sqrt` is defined as long as the eigenvalues are unique, and numerically determining this would be really unreliable.
---
### Author:  [@rafael.guerra](https://discourse.julialang.org/u/rafael.guerra)
#### Post date: [December 14, 2021, 3:40pm UTC](https://discourse.julialang.org/t/square-root-of-a-matrix-that-has-no-square-root/73081/3 "2021-12-14T15:40:45Z")
WolframAlpha produces the same result happily and quietly:
```nohighlight
MatrixPower[{{0, 1}, {0, 0}}, 1/2]
#result:
{{0, 0}, {0, 0}}
```
---
### Author:  [@stevengj](https://discourse.julialang.org/u/stevengj)
#### Post date: [December 14, 2021, 4:01pm UTC](https://discourse.julialang.org/t/square-root-of-a-matrix-that-has-no-square-root/73081/4 "2021-12-14T16:01:22Z")
> [@Oscar\_Smith](#):
>
> Throwing an error here would be hard. The `sqrt` is defined as long as the eigenvalues are unique, and numerically determining this would be really unreliable.
It’s fine to have repeated eigenvalues, even for a defective matrix:
```julia
julia> A = [4 1; 0 4]
2×2 Matrix{Int64}:
4 1
0 4
julia> S = sqrt(A)
2×2 Matrix{Float64}:
2.0 0.25
0.0 2.0
julia> S^2
2×2 Matrix{Float64}:
4.0 1.0
0.0 4.0
```
The problem only arises for repeated eigenvalues that are exactly zero and defective, I think?
It seems like we could check for this case in [`_sqrt_quasitriu_diag_block!`](https://github.com/JuliaLang/julia/blob/490d78cbfb8d3cc2bddabd34efbf34f3e1226a9b/stdlib/LinearAlgebra/src/triangular.jl#L2395-L2413) — perhaps you could just set a flag on [this line](https://github.com/JuliaLang/julia/blob/490d78cbfb8d3cc2bddabd34efbf34f3e1226a9b/stdlib/LinearAlgebra/src/triangular.jl#L2401) when `iszero(A[i,i])`, and throw an error if you hit that condition twice in a row (assuming the diagonal entries are sorted, as usual for Schur).
Of course, roundoff errors may defeat this too if the matrix is not in Schur-like form to start with:
```julia
julia> A = [0 1; 0 0]
2×2 Matrix{Int64}:
0 1
0 0
julia> sqrt(A) # fail
2×2 Matrix{Float64}:
0.0 0.0
0.0 0.0
julia> B = rand(2,2); # random change of basis
julia> A2 = B * A / B # similar to A, should also not have a square root
2×2 Matrix{Float64}:
0.68983 -0.154736
3.07534 -0.68983
julia> S = B \ sqrt(A2) * B # change basis from √A2 back to original basis
2×2 Matrix{ComplexF64}:
7.63631e-5+1.68025e-20im 6547.66+5.55286e-13im
-1.34717e-12-3.78013e-21im 7.63631e-5-4.38691e-13im
julia> S^2 # should == A if square root existed
2×2 Matrix{ComplexF64}:
-2.9895e-9-2.4751e-17im 1.0-2.8724e-9im
-2.05748e-16+1.36679e-26im -2.9895e-9-9.17507e-17im
julia> norm(S^2 - A) / norm(A) # loses half of the significant digits due to ill-conditioning
3.147765149816803e-8
```