Euphytica

, 214:51 | Cite as

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

Correction
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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)
Should read:

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)

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Agriculture, Food and WineThe University of AdelaideGlen OsmondAustralia
  2. 2.School of Mathematics and Applied Statistics, Faculty of Engineering and Information SciencesUniversity of WollongongWollongongAustralia

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