# rotational invariance

Started by April 24, 2010
```I am having difficulties to solve the following question. Would you please
help me?

suppose that X and Y are independent and each rotationally invariant on
R^k

a) Let X = X1 with Xi on R^k1 where i =1,2 and prove
X2
- Each Xi is rotationally invariant in its own right
- If Vi is in R[from K1 to K] with V'V = I, then V'X = X1

b) Let p denote any orthogonal projection with dim P = k1
determine the distribution of the correlation coefficient r=
X'PY/(|PX||PY|)

I think r is a special case of &sum;(Xi-barX)(Yi-barY) = X'PX
where P = I-n^(-1)11' but what should I do after?

```
```On 25 apr, 03:46, "dymin3" <dymin3@n_o_s_p_a_m.hotmail.com> wrote:
> I am having difficulties to solve the following question.

First of all: This is not a good place to go to have your homewrok
solved for you.

Second, it is very hard to figure head from tail in the mess you
came up with. I can *guess* that it has something to do with
linear algebra, but that's only a guess. As for the details,
there is far too little context to find out what this is about.

> I tried so hard to solve those two but i got stuck.

You don't understand the question - otherwise you would have
filled in the details missing. Which means you need to find

Rune
```
```Rune Allnor wrote:
> On 25 apr, 03:46, "dymin3" <dymin3@n_o_s_p_a_m.hotmail.com> wrote:
>> I am having difficulties to solve the following question.
>
> First of all: This is not a good place to go to have your homewrok
> solved for you.
>
> Second, it is very hard to figure head from tail in the mess you
> came up with. I can *guess* that it has something to do with
> linear algebra, but that's only a guess. As for the details,
> there is far too little context to find out what this is about.
>
>> I tried so hard to solve those two but i got stuck.
>
> You don't understand the question - otherwise you would have
> filled in the details missing. Which means you need to find
> your textbook on the subject and read very carfully. After you
>

meant "give me pointers", not "do my homework for me".  I mean --
usually the "do my homework" crowd adds "and make it snappy", without

Not all supervisors or tutors care, or are competent, or even manage one
out of the two.  So if he needs help, we should give him pointers (but
_not_ actually _do_ his homework for him).

--
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com
```
```dymin3 wrote:
> I am having difficulties to solve the following question. Would you please
> help me?
>
> suppose that X and Y are independent and each rotationally invariant on
> R^k
>
> a) Let X = X1 with Xi on R^k1 where i =1,2 and prove
>            X2
> - Each Xi is rotationally invariant in its own right
> - If Vi is in R[from K1 to K] with V'V = I, then V'X = X1
>
> b) Let p denote any orthogonal projection with dim P = k1
> determine the distribution of the correlation coefficient r=
> X'PY/(|PX||PY|)
>
> I think r is a special case of &sum;(Xi-barX)(Yi-barY) = X'PX
> where P = I-n^(-1)11' but what should I do after?
>
> I tried so hard to solve those two but i got stuck. Please help me T-T

Homework?  What's the subject?  If it's a problem out of a book, or a
class, mentioning the book or class title would go a long way toward us
knowing the subject domain to think in.

(and if it's linear algebra, be prepared for some of us to sit back,
watch and learn -- that's usually what I'm reduced to).

--
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com
```
```On 25 apr, 20:06, Tim Wescott <t...@seemywebsite.now> wrote:
> Rune Allnor wrote:
> > On 25 apr, 03:46, "dymin3" <dymin3@n_o_s_p_a_m.hotmail.com> wrote:
> >> I am having difficulties to solve the following question.
>
> > First of all: This is not a good place to go to have your homewrok
> > solved for you.
>
> > Second, it is very hard to figure head from tail in the mess you
> > came up with. I can *guess* that it has something to do with
> > linear algebra, but that's only a guess. As for the details,
> > there is far too little context to find out what this is about.
>
> >> I tried so hard to solve those two but i got stuck.
>
> > You don't understand the question - otherwise you would have
> > filled in the details missing. Which means you need to find
> > your textbook on the subject and read very carfully. After you
>
>
> meant "give me pointers", not "do my homework for me". &#2013266080;I mean --
> usually the "do my homework" crowd adds "and make it snappy", without
>
> Not all supervisors or tutors care, or are competent, or even manage one
> out of the two. &#2013266080;So if he needs help, we should give him pointers (but
> _not_ actually _do_ his homework for him).

OK, I'll grant the OP the benefit of the doubt.

*IF* this is about linear algebra (benevolent as I might be right
now, I still don't see exactly what he asks for), one needs to
first of all find out what a 'rotation' is. As far as I know, one
definition is 'a unitary matrix' U such that

UU' = U'U = I.

Next, we need to establish what property is invariant under
rotation. In the context of linear algebra the two most common
properties to watch out for are the norm and inner product (there
might be others, depending on what one is doing).

The whole exercise boils down to

1) Find the inner product <x,y> of the unrotated
vectors x and y
2) Find the inner product <Ux,Uy> of the rotated
versions of x and y
3) Show that <x,y> = <Ux,Uy> when U has the properties
of a rotation matrix, which means the inner product
is invariant under the rotation U.

Repeat 1-3 for the norms of the original vector x and the
rotated vector Ux to show that |x| = |Ux| (which is trivial
if one have already established the results for the inner
products).

Repeat 1-3 for whatever other properties one wants to study.

Rune
```