Notation in linear algebra and rule for square of matrix norm

[lishrimp]

Hi.

I have a few simple questions.

(<- sorry, please click this image.)

1. What does the notation in the red circle mean?

2. Is there a rule for expanding square of norm? (e.g. || A*B*C ||^2)
I don't really understand how the first eq. changes to the second eq.

Thanks. :)

 

[micromass]

Hi lishrimp!

Hi.

I have a few simple questions.

View attachment 37815 (<- sorry, please click this image.)

1. What does the notation in the red circle mean?

2. Is there a rule for expanding square of norm? (e.g. || A*B*C ||^2)
I don't really understand how the first eq. changes to the second eq.

Thanks. :)

My guess:

If $\Sigma$ is a matrix, then we can define a "norm" $\|~\|_\Sigma$ by setting

$$\|\mathbf{x}\|_\Sigma=\sqrt{\mathbf{x}^T\Sigma \mathbf{x}}$$

In your case, the matrix is $\Sigma^{-1}$, so the norm is

$$\|\mathbf{x}\|_{\Sigma^{-1}}=\sqrt{\mathbf{x}^T\Sigma^{-1} \mathbf{x}}$$

So

$$\|\mathbf{x}\|_{\Sigma^{-1}}^2=\mathbf{x}^T\Sigma^{-1} \mathbf{x}$$

So that equality is true by definition.

 

[lishrimp]

Thank you very much, micromass! :D