how to multiply out "transpose"?

**casperyc** · April 24th 2010, 08:22 AM

Hi

I am really confused with the "transpose" related stuff.

Can anyone work me through it?

and the paritial differentiation with respect to it

===========================
why is

$(Y-X\beta)^T(Y-X\beta)=Y^TY-2\beta^TX^TY^T+\beta^TX^TX\beta$

instead of

$(Y-X\beta)^T(Y-X\beta)<br /> =(Y^T-\beta^TX^T)(Y-X\beta)<br /> =Y^TY-Y^TX\beta-\beta^TX^TY+\beta^TX^TX\beta$

and differentiate with respect to beta is

$=-2X^TY+X^TX\beta$

**dwsmith** · April 24th 2010, 02:30 PM

Originally Posted by casperyc

Hi

I am really confused with the "transpose" related stuff.

Can anyone work me through it?

and the paritial differentiation with respect to it

If we are working with matrices, how do we know beta and beta hat commute? Or are we working with vector? What is beta?

$(\hat{\beta}-\beta)(\hat{\beta}^T-\beta^T)=\hat{\beta}\hat{\beta}^T-\hat{\beta}\beta^T-\beta\hat{\beta}^T+\beta\beta^T$

**casperyc** · April 24th 2010, 03:44 PM

Originally Posted by dwsmith

If we are working with matrices, how do we know beta and beta hat commute? Or are we working with vector? What is beta?

$(\hat{\beta}-\beta)(\hat{\beta}^T-\beta^T)=\hat{\beta}\hat{\beta}^T-\hat{\beta}\beta^T-\beta\hat{\beta}^T+\beta\beta^T$

Hi there,

I think what you have writtern is pretty obvious going to give me my answer.

So, my actual problem is dont understand the order of the multiplication,

say $(A-B)(A^T-B^T)$

what I would do is

$(A-B)(A^T-B^T)=A A^T - AB^T - B A^T + BB^T$

which give me a different 'middle' term...

$\beta$ is just a vector

and if you could see the second pic (which is just next to it)
it confuses me more....

**dwsmith** · April 24th 2010, 03:46 PM

Originally Posted by casperyc

Hi there,

I think what you have writtern is pretty obvious going to give me my answer.

So, my actual problem is dont understand the order of the multiplication,

say $(A-B)(A^T-B^T)$

what I would do is

$(A-B)(A^T-B^T)=A A^T - AB^T - B A^T + BB^T$

which give me a different 'middle' term...

$\beta$ is just a vector

Your multiplication is correct. You will receive two different middle terms. If we can show the terms commute, we could combine them.

**dwsmith** · April 24th 2010, 03:50 PM

If beta is a vector, $\beta\hat{\beta}^T$ is inner product.

**casperyc** · April 24th 2010, 03:54 PM

Originally Posted by dwsmith

If beta is a vector, $\beta\hat{\beta}^T$ is inner product.

Thanks.
I will think it through

Then what about the differentiation thing?
(which I have typed it out in the post)

**dwsmith** · April 24th 2010, 03:57 PM

Are the Xs and Ys matrices?

**casperyc** · April 24th 2010, 04:02 PM

Originally Posted by dwsmith

Are the Xs and Ys matrices?

just suppose

$y=n \times 1$

$X=n \times p$

$\beta=p \times 1$

**dwsmith** · April 24th 2010, 04:06 PM

Originally Posted by dwsmith

If we are working with matrices, how do we know beta and beta hat commute? Or are we working with vector? What is beta?

$(\hat{\beta}-\beta)(\hat{\beta}^T-\beta^T)=\hat{\beta}\hat{\beta}^T-\hat{\beta}\beta^T-\beta\hat{\beta}^T+\beta\beta^T$

Since these are vectors and we are dealing with inner product, $(\hat{\beta}-\beta)(\hat{\beta}^T-\beta^T)=\hat{\beta}\hat{\beta}^T-\hat{\beta}\beta^T-\beta\hat{\beta}^T+\beta\beta^T=\hat{\beta}\hat{\b eta}^T-2\hat{\beta}\beta^T+\beta\beta^T$

**dwsmith** · April 24th 2010, 04:20 PM

If in the second problem if beta is a scalar and X and Y are vectors, the quantity will simplify down to $Y^TY-2Y^TX\beta+X^TX\beta$

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