# Eigen() warntype

calling `LinearAlgebra.eigen()` gives warntype. Is it a bug?

``````import Statistics
import LinearAlgebra
function test(x::Matrix{Float64})
sigma = Statistics.cov(x)
eig = LinearAlgebra.eigen(sigma)
return eig
end

julia> @code_warntype test(randn(20, 5) )
Body::Union{Eigen{Complex{Float64},Complex{Float64},Array{Complex{Float64},2},Array{Complex{Float64},1}}, Eigen{Float64,Float64,Array{Float64,2},Array{Float64,1}}}
1 β %1 = Statistics.cov::Core.Compiler.Const(Statistics.cov, false)
β   %2 = invoke Statistics.:(#cov#37)(1::Int64, true::Bool, %1::Function, _2::Array{Float64,2})::Array{Float64,2}
β   %3 = LinearAlgebra.eigen::Core.Compiler.Const(LinearAlgebra.eigen, false)
β   %4 = invoke LinearAlgebra.:(#eigen#58)(true::Bool, true::Bool, %3::Function, %2::Array{Float64,2})::Union{Eigen{Complex{Float64},Complex{Float64},Array{Complex{Float64},2},Array{Complex{Float64},1}}, Eigen{Float64,Float64,Array{Float64,2},Array{Float64,1}}}
βββ      return %4
``````

The eigenvalues and eigenvectors can be real or complex. This is deliberate behaviour (not a bug).

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okay, even itβs not a bug, itβs a design issue: it gives a warntype, which would slows down runtime.

or, is it a special case that has no runtime impact?

moreover, as far as I know, for positive-semifinite matrix like the covariance, the Eigen values and vectors would always be real. Could we do better?

The amount of time spent on the warning will be tiny compared to the time taken to calculate the eigen-decomposition.

There is no type instability if the matrix is e.g. of type `Symmetric`.

thanks! so, the function should better be written as:

``````function test2(x::Matrix{Float64})
sigma = Statistics.cov(x)
eig = LinearAlgebra.eigen(LinearAlgebra.Symmetric(sigma) )
return eig
end

x = randn(20, 5)

julia> test(x) == test2(x)
true

@code_warntype test2(x)    # fine
``````
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Maybe `Statistics.cov` should automatically return a `Symmetric` matrix?

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But probably that would lead to type instability somewhere down the line.

Note that you can test that the result type of the calculation is the same as the type that Julia infers using

``````using Test
@inferred test2(x)
``````
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covariance is not only symmetric, itβs also positive-semi-finite (I think). It may be a big change to `LinearAlgebra` and `Statistics` to incorporate a special matrix type for it.

Yes, a covariance matrix is at least positive semi-definite.
If the number of linearly independent samples exceeds the dimensionality, it will be positive definite.

If youβre taking the eigendecomposition anyway, you can easily check whether all the eigenvalues are positive.

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