Question about hat matrix X(X'X)^(-1)X'

[julie94]

Hi everyone,

I am working on the following problem.

Suppose the set of vectors X1,..,Xk is a basis for linear space V1.
Suppose the set of vectors Y1,..,Yk is also a basis for linear space
V1.
Clearly the linear space spanned by the Xs equals the linear space
spanned by the Ys.

Construct an algebraic argument to show that
X(X'X)^(-1)X'=Y(Y'Y)^(-1)Y'

I am very confused, I am not sure what is meant by algebraic argument
in this instance, and I would welcome your ideas on how to tackle this
question.

Thanks in advance.

 

[HallsofIvy]

I am confused as to what is meant by "X" and "Y" here since you only mention, $\{X_k\}$ and $\{Y_k\}$ previously. But the reference to X', X^-1, Y', and Y^-1 indicate that X and Y are matrices or linear operators, not vectors. What is the relationship between X and $\{X_k\}$, between Y and $\{Y_k\}$?

If X and Y are matrices or linear operators then:
It looks to me that once you use (AB)^-1= B^-1A^-1 it becomes very easy to show that the two are the same!

 

[julie94]

Sorry I meant to write

X=[X1: X2 :...: Xk]
Y=[Y1: Y2 :...: Yk]

You are right, X and Y are matrices. Thanks a lot for the help.

 

[julie94]

I have tried (AB)-1= B-1A-1, but I am not getting what I need. Would you be kind enough to give me another hint.

 

[HallsofIvy]

You want to show that $X(X'X)^{-1}X'=Y(Y'Y)^{-1}Y'$.

Actually, you can show much more than that.

As I said, "$(AB)^{-1}= B^{-1}A^{-1}$ so that $(X'X)^{-1}= X^{-1}X'^{-1}$ and then $X(X'X)^{-1}X'= X(X^{-1}X'^{-1})X'$
Can you do that?

 

[julie94]

I understand the lines you wrote. But I do not know where to take it from here.

What should I do with
LaTeX Code: X(X^{-1}Xsingle-quote^{-1})Xsingle-quote
?
Thanks a lot for the help.

 

[julie94]

Do I need to write
X=PYP^{-1}
where P changes the basis from Y to X?
And plug in?