Euphytica

, 214:51

# Correction to: Comparison of a one- and two-stage mixed model analysis of Australia’s National Variety Trial Southern Region wheat data

Correction

## Correction to: Euphytica (2018) 214:44  https://doi.org/10.1007/s10681-018-2116-4

This article has been published with an erroneous version of Eq. 15. Please find the correct Eq. 15 in this document.

Derivation of (11)

Let T = [T1 T2] be an (n × n) non-singular transformation matrix such that T1 and T2, of dimension (n × t) and (n × (n − t)), satisfy
\begin{aligned}& \varvec{T}_{1}^{\mathsf {T}} \varvec{X} = \varvec{I}_{t} \varvec{T}_{2}^{\mathsf {T}} \hfill \\& \varvec{X} = {\mathbf{0}} \Leftrightarrow {\mathcal{R}} (\varvec{T}_{2} )\, \bot \,{\mathcal{R}} (\varvec{X}). \hfill \\ \end{aligned}
Likewise, let Q = [Q1 Q2] be an ((n − t) × (n − t)) non-singular transformation matrix such that Q1 and Q2, of dimension ((n − t) × d) and ((n – t) × (n – t − d)), satisfy
\begin{aligned}& \varvec{Q}_{1}^{\mathsf {T}} \varvec{T}_{2}^{\mathsf {T}} \varvec{X}_{g} = \varvec{I}_{d} \hfill \\& \varvec{Q}_{2}^{\mathsf {T}} \varvec{T}_{2}^{\mathsf {T}} \varvec{X}_{g} = {\mathbf{0}} \Leftrightarrow {\mathcal{R}} (\varvec{Q}_{2} )\, \bot\, {\mathcal{R}} (\varvec{T}_{2}^{\mathsf {T}} \varvec{X}_{g} ). \hfill \\ \end{aligned}
(15)

Derivation of (11)

Let T = [T1 T2] be an (n × n) non-singular transformation matrix such that T1 and T2, of dimension (n × t) and (n × (n − t)), satisfy
\begin{aligned} &\varvec{T}_{1}^{\mathsf {T}} \varvec{X} = \varvec{I}_{t} \hfill \\ &\varvec{T}_{2}^{\mathsf {T}} \varvec{X} = {\mathbf{0}} \Leftrightarrow {\mathcal{R}} (\varvec{T}_{2} )\, \bot\, {\mathcal{R}} (\varvec{X}). \hfill \\ \end{aligned}
Likewise, let Q = [Q1 Q2] be an ((n − t) × (n − t)) non-singular transformation matrix such that Q1 and Q2, of dimension ((n − t) × d) and ((n – t) × (n – t − d)), satisfy
\begin{aligned}& \varvec{Q}_{1}^{\mathsf {T}} \varvec{T}_{2}^{\mathsf {T}} \varvec{X}_{g} = \varvec{I}_{d} \hfill \\& \varvec{Q}_{2}^{\mathsf {T}} \varvec{T}_{2}^{\mathsf {T}} \varvec{X}_{g} = {\mathbf{0}} \Leftrightarrow {\mathcal{R}} (\varvec{Q}_{2} )\, \bot\, {\mathcal{R}} (\varvec{T}_{2}^{\mathsf {T}} \varvec{X}_{g} ). \hfill \\ \end{aligned}
(15)