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Exact(3)
The covariance structure of the genetic terms is var a D a S = C ⊗ A, where ⊗ is the Kronecker product of matrices, and C = σ A D 2 σ A DS σ A DS σ A S 2, where σ A D 2 is the direct genetic variance, σ A S 2 is the indirect genetic variance, and σ A DS is the direct indirect genetic covariance.
Proof: Let A and B are two stochastic matrices, and C = A.B is their product.
Where (a) and (b) correspond to the case of full covariance matrices and, (c) and (d) correspond to the case of diagonal covariance matrices.
Similar(57)
A and B are substitutional elements in the matrix and C is an interstitial component, assumed to preferentially bind to B atoms.
which reduce to (17), (18) if A is the unit matrix and c = 0.
where A j is a square semi-definite matrix and c j is a real constant.
where I is the N×N identity matrix and c is the Welch bound given by (3).
Assume a COW matrix of size (64, 104) with the signature matrix D = H8 ⊗ C8 × 13 (H8 is an 8 × 8 Hadamard matrix and C is a COW matrix).
Let A be a regular (ntimes n) matrix, B be an (ntimes q) matrix, and C be a regular (qtimes q) matrix.
Let A be a regular n × n matrix, B be an n × q matrix, and C be a q × q regular matrix.
Let A be a regular n × n matrix, D be a q × q matrix, B be an n × q matrix, and C be a q × n matrix, and the matrix M = ( A B C D ). be regular.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com