Hi MHF,

Consider 2 vectors x and m with the same dimension. Is the following condition true?

$\displaystyle (x - m)(x - m)^T == xx^T - mm^T$

It has been a while since I did linear algebra so some help is much appreciated!

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- May 21st 2011, 02:25 PMikkuhMatrix (vector) multiplication
Hi MHF,

Consider 2 vectors x and m with the same dimension. Is the following condition true?

$\displaystyle (x - m)(x - m)^T == xx^T - mm^T$

It has been a while since I did linear algebra so some help is much appreciated! - May 21st 2011, 03:00 PMpickslides
It is true. As x and m have the same dimension say pxq, simply multiply out each side of your equation, you should get pxp in both cases.

- May 21st 2011, 03:14 PMPlato
- May 21st 2011, 03:43 PMpickslides
The dimensions of the solutions may be the same, maybe not the solution itself.

- May 22nd 2011, 04:13 PMikkuh
That is exactly what I was thinking. Unfortunately problem I need to solve is this:

$\displaystyle S = \frac{1}{n-1}\sum\limits_{i=1}^n ({\bf x}_i - {\bf m})({\bf x}_i - {\bf m})^T$ where

$\displaystyle {\bf m} = \frac{1}{n}\sum\limits_{i=1}^n x_i$

Prove that the expression for the covariance matrix (above) can be rewritten as:

$\displaystyle S = \frac{(\sum\limits_{i=1}^n {\bf x}_i {\bf x}_i^T) - n {\bf m}{\bf m}^t }{n-1}$

I might have split the question wrong before. Anyone who sees now how you can rewrite the expression?