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Supplementary Methods I:
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Proof of the rotation invariance property of the genomic heritability
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We show here that the genetic and genomic variances (expressions 3 and 6, respectively) are invariant under
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linear transformations of genotypes; therefore, these parameters, and functions thereof such as the trait
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heritability and the genomic heritability, do not depend on the way genotypes are coded. To see this consider
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arbitrary linear transformations (recoding is a full-rank linear transformation) of the QTL and marker
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genotypes of the form: 𝑧̃𝑖 = 𝜏𝑧 + 𝑇𝑧 𝑧𝑖 and 𝑥̃𝑖 = 𝜏𝑥 + 𝑇𝑥 𝑥𝑖 . After transformation, the relevant covariance
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matrices become 𝛴𝑧̃ = 𝑇𝑧 𝛴𝑧 𝑇𝑧 ′, 𝛴𝑥̃ = 𝑇𝑥 𝛴𝑥 𝑇𝑥′ , 𝛴𝑧̃𝑔 = 𝑇𝑧 𝛴𝑧𝑔 , 𝛴𝑥̃𝑧 = 𝑇𝑥 𝛴𝑥𝑧 and 𝛴𝑥̃𝑧̃ = 𝑇𝑥̃ 𝛴𝑥𝑧 𝑇𝑧̃ ′. Therefore,
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the effects of the transformed QTL and marker genotypes become
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′−1 −1 −1
𝛼̃ = Σ𝑧−1
Σ𝑧 𝑇𝑧 𝑇𝑧 Σ𝑧𝑔 = 𝑇𝑧 ′−1 Σ𝑧−1 Σ𝑧𝑔 = 𝑇𝑧 ′−1 α
̃ Σ𝑧̃𝑔 = 𝑇𝑧
and
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𝛽̃ = 𝑉𝑎𝑟(𝑥̃𝑖 )−1 𝐶𝑜𝑣(𝑥̃𝑖 , 𝑧̃𝑖 ′𝛼̃) = (𝑇𝑥 ′−1 Σ𝑥−1 𝑇𝑥 −1 ) (𝑇𝑥 Σ𝑥𝑧 𝑇𝑧̃ ′) 𝑇𝑧 ′−1 α = 𝑇𝑥 ′−1 Σ𝑥−1 Σ𝑥𝑧 α = 𝑇𝑥 ′−1 𝛽 ,
respectively. Therefore, the additive and genomic variance become
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𝑉𝑎𝑟(𝛼̃ ′ 𝑧̃𝑖 ) = α′𝑇𝑧
−1
𝑇𝑧 Σ𝑧 𝑇𝑧 ′𝑇𝑧
′−1
α = α′ Σ𝑧 α
and
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𝑉𝑎𝑟(𝛽̃ ′𝑥̃𝑖 ) = 𝛽′𝑇𝑥 −1 𝑇𝑥 𝛴𝑥 𝑇𝑥′ 𝑇𝑥 ′−1 𝛽= 𝛽′𝛴𝑥 𝛽
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This means that while the sign of the effects is arbitrary (i.e., it depends on how genotypes are coded), the
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variance parameters are invariant with respect to any arbitrary linear transformation, including recoding.
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