- #1
EdMel
- 13
- 0
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, \mathbf{y} and \tilde{\beta} are a mx1 vectors, and \mathbf{X} is a mxn matrix.
I understand the equation
(\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}<br />
, but then it is stated
\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)}
, and I do not understand why -\mathbf{y}^{\text{T}}\mathbf{X}\tilde{\beta}=-\tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{y} in equation (1).
I understand the transpose identity (\mathbf{y}^{\text{T}}\tilde{\beta}\mathbf{X})^{\text{T}}= \mathbf{X}^{\text{T}}\tilde{\beta}^{\text{T}}\mathbf{y},
but then (1) would be
\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},
and (1) would only be true if \tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{y} is s symmetric matrix, which I think it need not be.
What am I missing here?
Thanks in advance,
Ed
The following equations are from linear regression model notes but there is an aspect of the matrix algebra I do not get.
I have, \mathbf{y} and \tilde{\beta} are a mx1 vectors, and \mathbf{X} is a mxn matrix.
I understand the equation
(\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}<br />
, but then it is stated
\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)}
, and I do not understand why -\mathbf{y}^{\text{T}}\mathbf{X}\tilde{\beta}=-\tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{y} in equation (1).
I understand the transpose identity (\mathbf{y}^{\text{T}}\tilde{\beta}\mathbf{X})^{\text{T}}= \mathbf{X}^{\text{T}}\tilde{\beta}^{\text{T}}\mathbf{y},
but then (1) would be
\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},
and (1) would only be true if \tilde{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{y} is s symmetric matrix, which I think it need not be.
What am I missing here?
Thanks in advance,
Ed