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Transpose of the product of matrices problem

  1. Aug 30, 2013 #1
    Hi,

    The following equations are from linear regression model notes but there is an aspect of the matrix algebra I do not get.

    I have, [itex]\mathbf{y}[/itex] and [itex]\tilde{\beta}[/itex] are a mx1 vectors, and [itex] \mathbf{X}[/itex] is a mxn matrix.

    I understand the equation
    [tex](\mathbf{y}-\mathbf{X}\tilde{\beta})^{\text{T}}(\mathbf{y}-\mathbf{X}\tilde{\beta})= \mathbf{y}^{\text{T}}\mathbf{y}-\tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{y}-\mathbf{y}^{\text{T}}\mathbf{X}\tilde{\beta}+ \tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{X}\tilde{\beta}
    [/tex]
    , but then it is stated
    [tex]\mathbf{y}^{\text{T}}\mathbf{y}-\tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{y}-\mathbf{y}^{\text{T}}\mathbf{X}\tilde{\beta}-\tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{X}\tilde{\beta}= \mathbf{y}^{\text{T}}\mathbf{y}-2\tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{y}+\tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{X}\tilde{\beta}\qquad\text{(1)}[/tex]
    , and I do not understand why [itex]-\mathbf{y}^{\text{T}}\mathbf{X}\tilde{\beta}=-\tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{y}[/itex] in equation (1).

    I understand the transpose identity [itex](\mathbf{y}^{\text{T}}\tilde{\beta}\mathbf{X})^{\text{T}}= \mathbf{X}^{\text{T}}\tilde{\beta}^{\text{T}}\mathbf{y}[/itex],
    but then (1) would be
    [tex]\mathbf{y}^{\text{T}}\mathbf{y}-\tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{y}-\mathbf{y}^{\text{T}}\mathbf{X}\tilde{\beta}-\tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{X}\tilde{\beta}= \mathbf{y}^{\text{T}}\mathbf{y}-\tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{y}-(\tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{y})^{\text{T}}+ \tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{X}\tilde{\beta}[/tex],
    and (1) would only be true if [itex]\tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{y}[/itex] is s symmetric matrix, which I think it need not be.

    What am I missing here?

    Thanks in advance,

    Ed
     
  2. jcsd
  3. Aug 30, 2013 #2

    chiro

    User Avatar
    Science Advisor

    Hey EdMel.

    Hint: Is the quantity a scalar? (If it is then what does this imply about the appropriate transpose?)
     
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