Transpose of the product of matrices problem

  • Thread starter EdMel
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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
 

Answers and Replies

  • #2
chiro
Science Advisor
4,790
131
Hey EdMel.

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

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