title: Interpretation of a multivariate linear model
body: <p>I have a question about the model matrix and parameter matrix of the linear multivariate model fit to the iris data set in R. This is the code (function lm) I am using and the outputs:</p>
  <pre><code>&gt; (fm = lm(cbind(iris[,1],iris[,2],iris[,3],iris[,4])~(iris[,5])))
  
  Call:
  lm(formula = cbind(iris[, 1], iris[, 2], iris[, 3], iris[, 4]) ~ 
      (iris[, 5]))
  
  Coefficients:
                       [,1]    [,2]    [,3]    [,4]  
  (Intercept)           5.006   3.428   1.462   0.246
  iris[, 5]versicolor   0.930  -0.658   2.798   1.080
  iris[, 5]virginica    1.582  -0.454   4.090   1.780
  </code></pre>
  <p>My objective is to get the decomposition of the model in matrices:</p>
  <p><strong>The model matrix A and the 4 by 4 parameter matrix β for the multivariate model
  X = Aβ + E</strong></p>
  <p>From my knowledge the matrix β is the Coefficient matrix just obtained above:</p>
  <pre><code>                    [,1]    [,2]    [,3]    [,4]  
  (Intercept)           5.006   3.428   1.462   0.246
  iris[, 5]versicolor   0.930  -0.658   2.798   1.080
  iris[, 5]virginica    1.582  -0.454   4.090   1.780
  </code></pre>
  <p>Therefore, is not the matrix  β a 3 by 4 matrix (3 rows and 4 columns) instead of a 4 x 4 matrix?</p>