Matrix proof help PLZ!

twinwings · **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?

Shadow · **Posted:** Fri, 30 Sep 2011 23:44:30 UTC

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 $P_x=X(X^\mathrm{T}X)^1X^\mathrm{T}$ "?

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?

twinwings · **Joined:** Fri, 30 Sep 2011 22:14:42 UTC **Posts:** 8

Shadow 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 $P_x=X(X^\mathrm{T}X)^1X^\mathrm{T}$ "?

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?

Shadow · **Posted:** Fri, 30 Sep 2011 23:59:53 UTC

twinwings wrote:

Shadow 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 $P_x=X(X^\mathrm{T}X)^1X^\mathrm{T}$ "?

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 $P=\begin{pmatrix} 1 & 0\end{pmatrix}:\mathbb{R}^2\to\mathbb{R}^1$ is a projection which is not square, despite the fact that PP^T=1=I_1

is an invertible square matrix, just as P^TP=I_2

.

I also already got that P_x

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.

twinwings · **Joined:** Fri, 30 Sep 2011 22:14:42 UTC **Posts:** 8

Shadow wrote:

twinwings wrote:

Shadow 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 $P_x=X(X^\mathrm{T}X)^1X^\mathrm{T}$ "?

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 $P=\begin{pmatrix} 1 & 0\end{pmatrix}:\mathbb{R}^2\to\mathbb{R}^1$ is a projection which is not square, despite the fact that PP^T=1=I_1

is an invertible square matrix, just as P^TP=I_2

.

I also already got that P_x

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?

Shadow · **Posted:** Sat, 1 Oct 2011 00:25:46 UTC

twinwings wrote:

Shadow wrote:

twinwings wrote:

Shadow 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 $P_x=X(X^\mathrm{T}X)^1X^\mathrm{T}$ "?

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 $P=\begin{pmatrix} 1 & 0\end{pmatrix}:\mathbb{R}^2\to\mathbb{R}^1$ is a projection which is not square, despite the fact that PP^T=1=I_1

is an invertible square matrix, just as P^TP=I_2

.

I also already got that P_x

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?

twinwings · **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

Shadow · **Posted:** Sat, 1 Oct 2011 00:35:16 UTC

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 XX^T

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

twinwings · **Joined:** Fri, 30 Sep 2011 22:14:42 UTC **Posts:** 8

Shadow 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 XX^T

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

Shadow · **Posted:** Sat, 1 Oct 2011 00:43:53 UTC

twinwings wrote:

Shadow 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 XX^T

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 X=X^T=0_n

the nxn zero matrix? You CANNOT conclude XX^T

is the identity for arbitrary X, in particular you don't know that XX^T

is invertible.

twinwings · **Joined:** Fri, 30 Sep 2011 22:14:42 UTC **Posts:** 8

Shadow wrote:

twinwings wrote:

Shadow 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 XX^T

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 X=X^T=0_n

the nxn zero matrix? You CANNOT conclude XX^T

is the identity for arbitrary X, in particular you don't know that XX^T

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

Shadow · **Posted:** Sat, 1 Oct 2011 00:52:58 UTC

twinwings wrote:

Shadow wrote:

twinwings wrote:

Shadow 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 XX^T

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 X=X^T=0_n

the nxn zero matrix? You CANNOT conclude XX^T

is the identity for arbitrary X, in particular you don't know that XX^T

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 $a^{-1}$ exists, nothing works.

twinwings · **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

twinwings · **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

Shadow · **Posted:** Sat, 1 Oct 2011 01:23:05 UTC

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 P_XP_X=P_X

. To see what the dimensions of P_X

are, it's just looking at the dimensions of the factors, and you'll find you get n x n.

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