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 Post subject: Matrix proof help PLZ!Posted: Fri, 30 Sep 2011 22:22:45 UTC

Joined: Fri, 30 Sep 2011 22:14:42 UTC
Posts: 8
Hey there,

This is my first post and I was wondering whether any of you can help me out?

Consider the n x k matrix X, and the projection matrix Px = X ((XTX)^1) XT

where XT= is the X transpose and Px is a term on its own NOT P*x

a) What are the dimensions of Px, (number of rows and columns)?

b) Prove PxPx = Px.

c) Does Py = Px if Y = XA, where A is an k x k nonsingular matrix?

I was able to solve a) and b)...I think...dimensions of Px=nxn and Px=I (I still have some trouble with this)

However I am having trouble solving c)

Can anyone help?

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 Post subject: Re: Matrix proof help PLZ!Posted: Fri, 30 Sep 2011 23:44:30 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
twinwings wrote:
Hey there,

This is my first post and I was wondering whether any of you can help me out?

Consider the n x k matrix X, and the projection matrix Px = X ((XTX)^1) XT

where XT= is the X transpose and Px is a term on its own NOT P*x

a) What are the dimensions of Px, (number of rows and columns)?

b) Prove PxPx = Px.

c) Does Py = Px if Y = XA, where A is an k x k nonsingular matrix?

I was able to solve a) and b)...I think...dimensions of Px=nxn and Px=I (I still have some trouble with this)

However I am having trouble solving c)

Can anyone help?

I'm having a bit of trouble reading what you typed, is this accurate:

"Let "?

Why is there a to the 1 power in there? What are the lower-case x and y? How are they connected to big X and Y?

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 Post subject: Re: Matrix proof help PLZ!Posted: Fri, 30 Sep 2011 23:55:57 UTC

Joined: Fri, 30 Sep 2011 22:14:42 UTC
Posts: 8
twinwings wrote:
Hey there,

This is my first post and I was wondering whether any of you can help me out?

Consider the n x k matrix X, and the projection matrix Px = X ((XTX)^1) XT

where XT= is the X transpose and Px is a term on its own NOT P*x

a) What are the dimensions of Px, (number of rows and columns)?

b) Prove PxPx = Px.

c) Does Py = Px if Y = XA, where A is an k x k nonsingular matrix?

I was able to solve a) and b)...I think...dimensions of Px=nxn and Px=I (I still have some trouble with this)

However I am having trouble solving c)

Can anyone help?

I'm having a bit of trouble reading what you typed, is this accurate:

"Let "?

Why is there a to the 1 power in there? What are the lower-case x and y? How are they connected to big X and Y?

Hi and thanks,

the bracket is to the power of -1. and according to the question, Px is defined as the Projection matrix. Px is a term in itself, it is not P * x . As for relation between small x and y, I can't seem to figure out anything else past what the question mentions?

do you think it's correct if I say

if (XT,X)^-1 exists,
then the product of X transpose times X has an existing inverse,

hence det(XT*X) does not equal 0,
and that also means det(XT) and det(X) do not equal zero
so therefore XT and X are square matrices, are invertible and their dimensions nxk, is actually, nxk where n=k?

is this correct?

Top

 Post subject: Re: Matrix proof help PLZ!Posted: Fri, 30 Sep 2011 23:59:53 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
twinwings wrote:
twinwings wrote:
Hey there,

This is my first post and I was wondering whether any of you can help me out?

Consider the n x k matrix X, and the projection matrix Px = X ((XTX)^1) XT

where XT= is the X transpose and Px is a term on its own NOT P*x

a) What are the dimensions of Px, (number of rows and columns)?

b) Prove PxPx = Px.

c) Does Py = Px if Y = XA, where A is an k x k nonsingular matrix?

I was able to solve a) and b)...I think...dimensions of Px=nxn and Px=I (I still have some trouble with this)

However I am having trouble solving c)

Can anyone help?

I'm having a bit of trouble reading what you typed, is this accurate:

"Let "?

Why is there a to the 1 power in there? What are the lower-case x and y? How are they connected to big X and Y?

Hi and thanks,

the bracket is to the power of -1. and according to the question, Px is defined as the Projection matrix. Px is a term in itself, it is not P * x . As for relation between small x and y, I can't seem to figure out anything else past what the question mentions?

do you think it's correct if I say

if (XT,X)^-1 exists,
then the product of X transpose times X has an existing inverse,

hence det(XT*X) does not equal 0,
and that also means det(XT) and det(X) do not equal zero
so therefore XT and X are square matrices, are invertible and their dimensions nxk, is actually, nxk where n=k?

is this correct?

No, this is not true. Note that the matrix is a projection which is not square, despite the fact that is an invertible square matrix, just as .

I also already got that was a term by itself, that's not important, but unless x and y have some connection to X and Y, then this question doesn't make sense.

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 Post subject: Re: Matrix proof help PLZ!Posted: Sat, 1 Oct 2011 00:06:20 UTC

Joined: Fri, 30 Sep 2011 22:14:42 UTC
Posts: 8
twinwings wrote:
twinwings wrote:
Hey there,

This is my first post and I was wondering whether any of you can help me out?

Consider the n x k matrix X, and the projection matrix Px = X ((XTX)^1) XT

where XT= is the X transpose and Px is a term on its own NOT P*x

a) What are the dimensions of Px, (number of rows and columns)?

b) Prove PxPx = Px.

c) Does Py = Px if Y = XA, where A is an k x k nonsingular matrix?

I was able to solve a) and b)...I think...dimensions of Px=nxn and Px=I (I still have some trouble with this)

However I am having trouble solving c)

Can anyone help?

I'm having a bit of trouble reading what you typed, is this accurate:

"Let "?

Why is there a to the 1 power in there? What are the lower-case x and y? How are they connected to big X and Y?

Hi and thanks,

the bracket is to the power of -1. and according to the question, Px is defined as the Projection matrix. Px is a term in itself, it is not P * x . As for relation between small x and y, I can't seem to figure out anything else past what the question mentions?

do you think it's correct if I say

if (XT,X)^-1 exists,
then the product of X transpose times X has an existing inverse,

hence det(XT*X) does not equal 0,
and that also means det(XT) and det(X) do not equal zero
so therefore XT and X are square matrices, are invertible and their dimensions nxk, is actually, nxk where n=k?

is this correct?

No, this is not true. Note that the matrix is a projection which is not square, despite the fact that is an invertible square matrix, just as .

I also already got that was a term by itself, that's not important, but unless x and y have some connection to X and Y, then this question doesn't make sense.

Darn it,

Well, I posed the exact question from the handout. The only thing that I added to it was that Px is a term on its own.

This is actually a question from Econometrics (Intro). You think I should contact the prof?

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 Post subject: Re: Matrix proof help PLZ!Posted: Sat, 1 Oct 2011 00:25:46 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
twinwings wrote:
twinwings wrote:
twinwings wrote:
Hey there,

This is my first post and I was wondering whether any of you can help me out?

Consider the n x k matrix X, and the projection matrix Px = X ((XTX)^1) XT

where XT= is the X transpose and Px is a term on its own NOT P*x

a) What are the dimensions of Px, (number of rows and columns)?

b) Prove PxPx = Px.

c) Does Py = Px if Y = XA, where A is an k x k nonsingular matrix?

I was able to solve a) and b)...I think...dimensions of Px=nxn and Px=I (I still have some trouble with this)

However I am having trouble solving c)

Can anyone help?

I'm having a bit of trouble reading what you typed, is this accurate:

"Let "?

Why is there a to the 1 power in there? What are the lower-case x and y? How are they connected to big X and Y?

Hi and thanks,

the bracket is to the power of -1. and according to the question, Px is defined as the Projection matrix. Px is a term in itself, it is not P * x . As for relation between small x and y, I can't seem to figure out anything else past what the question mentions?

do you think it's correct if I say

if (XT,X)^-1 exists,
then the product of X transpose times X has an existing inverse,

hence det(XT*X) does not equal 0,
and that also means det(XT) and det(X) do not equal zero
so therefore XT and X are square matrices, are invertible and their dimensions nxk, is actually, nxk where n=k?

is this correct?

No, this is not true. Note that the matrix is a projection which is not square, despite the fact that is an invertible square matrix, just as .

I also already got that was a term by itself, that's not important, but unless x and y have some connection to X and Y, then this question doesn't make sense.

Darn it,

Well, I posed the exact question from the handout. The only thing that I added to it was that Px is a term on its own.

This is actually a question from Econometrics (Intro). You think I should contact the prof?

Are X or x or Y or y defined on the handout at all? Perhaps you said something about the notation in class?

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 Post subject: Re: Matrix proof help PLZ!Posted: Sat, 1 Oct 2011 00:32:04 UTC

Joined: Fri, 30 Sep 2011 22:14:42 UTC
Posts: 8
Nothing my friend,

the question is exactly as you see it.

I proved that Px=PxPx because I just multiply Px by itself and get the same thing since the middle becomes the identity.

However, c) is quite confusing

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 Post subject: Re: Matrix proof help PLZ!Posted: Sat, 1 Oct 2011 00:35:16 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
twinwings wrote:
Nothing my friend,

the question is exactly as you see it.

I proved that Px=PxPx because I just multiply Px by itself and get the same thing since the middle becomes the identity.

However, c) is quite confusing

Hold it, how do you know that is an identity matrix? You didn't say that was given, and it's certainly not true in general.

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 Post subject: Re: Matrix proof help PLZ!Posted: Sat, 1 Oct 2011 00:39:56 UTC

Joined: Fri, 30 Sep 2011 22:14:42 UTC
Posts: 8
twinwings wrote:
Nothing my friend,

the question is exactly as you see it.

I proved that Px=PxPx because I just multiply Px by itself and get the same thing since the middle becomes the identity.

However, c) is quite confusing

Hold it, how do you know that is an identity matrix? You didn't say that was given, and it's certainly not true in general.

Lets see,

Px=PxPx
(X((XTX)^-1)XT) (X((XTX)^-1)XT)

By associate property
X*((XTX)^-1)(XTX)((XTX)^-1)XT

Hence ((XTX)^-1)(XTX)=I

so

X*I*((XTX)^-1)XT

which is also equal to Px

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 Post subject: Re: Matrix proof help PLZ!Posted: Sat, 1 Oct 2011 00:43:53 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
twinwings wrote:
twinwings wrote:
Nothing my friend,

the question is exactly as you see it.

I proved that Px=PxPx because I just multiply Px by itself and get the same thing since the middle becomes the identity.

However, c) is quite confusing

Hold it, how do you know that is an identity matrix? You didn't say that was given, and it's certainly not true in general.

Lets see,

Px=PxPx
(X((XTX)^-1)XT) (X((XTX)^-1)XT)

By associate property
X*((XTX)^-1)(XTX)((XTX)^-1)XT

Hence ((XTX)^-1)(XTX)=I

so

X*I*((XTX)^-1)XT

which is also equal to Px

You haven't answered my question, what if the nxn zero matrix? You CANNOT conclude is the identity for arbitrary X, in particular you don't know that is invertible.

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 Post subject: Re: Matrix proof help PLZ!Posted: Sat, 1 Oct 2011 00:49:05 UTC

Joined: Fri, 30 Sep 2011 22:14:42 UTC
Posts: 8
twinwings wrote:
twinwings wrote:
Nothing my friend,

the question is exactly as you see it.

I proved that Px=PxPx because I just multiply Px by itself and get the same thing since the middle becomes the identity.

However, c) is quite confusing

Hold it, how do you know that is an identity matrix? You didn't say that was given, and it's certainly not true in general.

Lets see,

Px=PxPx
(X((XTX)^-1)XT) (X((XTX)^-1)XT)

By associate property
X*((XTX)^-1)(XTX)((XTX)^-1)XT

Hence ((XTX)^-1)(XTX)=I

so

X*I*((XTX)^-1)XT

which is also equal to Px

You haven't answered my question, what if the nxn zero matrix? You CANNOT conclude is the identity for arbitrary X, in particular you don't know that is invertible.

The way I see it suppose,

a=XTX
then a^-1= is the inverse of XTX

so (a times a^-1)= I

this scenario occurs in the middle of the multiplication Px * Px. I never actually said or assumed any of the things above

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 Post subject: Re: Matrix proof help PLZ!Posted: Sat, 1 Oct 2011 00:52:58 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
twinwings wrote:
twinwings wrote:
twinwings wrote:
Nothing my friend,

the question is exactly as you see it.

I proved that Px=PxPx because I just multiply Px by itself and get the same thing since the middle becomes the identity.

However, c) is quite confusing

Hold it, how do you know that is an identity matrix? You didn't say that was given, and it's certainly not true in general.

Lets see,

Px=PxPx
(X((XTX)^-1)XT) (X((XTX)^-1)XT)

By associate property
X*((XTX)^-1)(XTX)((XTX)^-1)XT

Hence ((XTX)^-1)(XTX)=I

so

X*I*((XTX)^-1)XT

which is also equal to Px

You haven't answered my question, what if the nxn zero matrix? You CANNOT conclude is the identity for arbitrary X, in particular you don't know that is invertible.

The way I see it suppose,

a=XTX
then a^-1= is the inverse of XTX

so (a times a^-1)= I

this scenario occurs in the middle of the multiplication Px * Px. I never actually said or assumed any of the things above

Yes, but what if a doesn't have an inverse? Without the assumption that exists, nothing works.

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 Post subject: Re: Matrix proof help PLZ!Posted: Sat, 1 Oct 2011 00:54:15 UTC

Joined: Fri, 30 Sep 2011 22:14:42 UTC
Posts: 8
Px=PxPx
(X((XTX)^-1)XT) (X((XTX)^-1)XT)

By associate property
X*((XTX)^-1)(XTX)((XTX)^-1)XT

Hence ((XTX)^-1)(XTX)=I

so

X*I*((XTX)^-1)XT

which is also equal to Px

I never assumed XTX=XXT

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 Post subject: Re: Matrix proof help PLZ!Posted: Sat, 1 Oct 2011 00:56:36 UTC

Joined: Fri, 30 Sep 2011 22:14:42 UTC
Posts: 8
The reason I believe (X^T * X)^-1, DOES exist is because the professor states the equation.

Why would the prof say Px= X * something that doesn't exist * X^T

and I can't even bring the inverse inside it because X^T or X are not square

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 Post subject: Re: Matrix proof help PLZ!Posted: Sat, 1 Oct 2011 01:23:05 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
twinwings wrote:
The reason I believe (X^T * X)^-1, DOES exist is because the professor states the equation.

Why would the prof say Px= X * something that doesn't exist * X^T

and I can't even bring the inverse inside it because X^T or X are not square

Ah, OK, so there is an implicit assumption. That was a mistake on the professor's part then, he should have said that. I'm willing to bet y should be Y and x should be X, since it seems he is defining P through X.

From this I can verify the correctness of your proof that . To see what the dimensions of are, it's just looking at the dimensions of the factors, and you'll find you get n x n.

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