# matrix trace and idempotent

• Oct 14th 2010, 06:36 PM
rainer
matrix trace and idempotent
\$\displaystyle X(X'X)^{-1}X'\$

X is an n x K non-singular full rank matrix where n > K

Is the result of this matrix operation an idempotent matrix? (It seems to be when I do it in stata.)

Given the properties below, I am told it is possible to express the trace of the matrix that results from the above operation in terms of n and K. But I have been unable to figure out how. Can someone give me a clue?

\$\displaystyle (A')^{-1}=(A^{-1})'\$

\$\displaystyle tr(ABC)=tr(CAB)\$

(where A,B, and C are non-singular square matrices)
• Oct 14th 2010, 07:27 PM
tonio
Quote:

Originally Posted by rainer
\$\displaystyle X(X'X)^{-1}X'\$

X is an n x K non-singular full rank matrix where n > K

Is the result of this matrix operation an idempotent matrix? (It seems to be when I do it in stata.)

Given the properties below, I am told it is possible to express the trace of the matrix that results from the above operation in terms of n and K. But I have been unable to figure out how. Can someone give me a clue?

\$\displaystyle (A')^{-1}=(A^{-1})'\$

\$\displaystyle tr(ABC)=tr(CAB)\$

(where A,B, and C are non-singular square matrices)

What is \$\displaystyle X'\$ , for a square matrix??

Tonio
• Oct 14th 2010, 08:11 PM
rainer
Quote:

Originally Posted by tonio
What is \$\displaystyle X'\$ , for a square matrix??

Tonio

But X is not square. It is n-by-K, where n>K.
• Oct 14th 2010, 11:10 PM
tonio
Quote:

Originally Posted by rainer
But X is not square. It is n-by-K, where n>K.

Whatever: what is \$\displaystyle X'\$ , anyway?

Tonio
• Oct 15th 2010, 12:10 AM
rainer
Quote:

Originally Posted by tonio
Whatever: what is \$\displaystyle X'\$ , anyway?

Tonio

\$\displaystyle X'\$ is the transpose of \$\displaystyle X\$.

\$\displaystyle X'X\$ yields a square matrix. I think that's part of the trick.