# orthogonal matrices vs. orthogonal columns

I'm just reading a book on econometrics and now I'm stuck with a problem:

There is a Theorem on "Orthogonal Partitioned Regression" which says:

"In the multiple linear least squares regression of $y$ on two sets of variables $X_1$ and $X_2$, if the two sets of variables are orthogonal, then the separate coefficient vectors can be obtained by separate regressions of $y$ on $X_1$ alone and $y$ on $X_2$ alone. ..."

Furthermore, the authors says that $X_1$ and $X_2$ are orthogonal if: $$(X_1)' X_2=0 \, ,$$ which confuses me because so far I always thought the condition for orthogonality between two matrices is that $(X_1)' X_2=I$.

The book is: W. Greene, Econometric Analysis, 7th ed. / Theorem 3.1 / p.33

What I found so far is:

Link 1 and Link 2 (search for: "imagine"), but neither really helps me with my problem.

BR Fabian

In the sentence beginning with "Imagine" sentence in your Link 2, I think this is the definition of "orthogonal matrices" that you want: Two matrices $X_{1}, X_{2}$ are orthogonal provided that "the columns of $X_{1}$ are orthogonal to the columns of $X_{2}$ such that $X_{1}'X_{2} = 0$."

I have not heard of a notion of "orthogonal matrices" outside of this question. An orthogonal matrix is a matrix $X$ satisfying $X'X = I$.

• Thank you for your answer. Indeed, the author later on specifies and says: "... if the columns of X are orthogonal ..." and therefore (X1)'(X2)=0 is fine. However, according to the definition of wikipedia the definition of orthogonality is (X1)'(X2)=I. So I wonder what is the difference between two matrices being orthogonal and the columns of two matrices being ortohogonal. – Fabian Aug 19 '14 at 17:46
• @Fabian No, that's not quite what Wikipedia says. Note that there's only a single matrix in the defining equation $Q^T Q = I$ in the article you linked. In your definition, there are 2 different matrices. – André 3000 Aug 19 '14 at 18:48

Two vectors $v,w \in \mathbb{R}^n$ are orthogonal iff $v^t w = 0$ where $t$ indicates the transpose. Really, we're using the dot product given by $\langle v, w \rangle = v^t w$.

There is a different notion of orthogonality for matrices. Here's one definition of an orthogonal matrix: $O \in \rm{M}_n(\mathbb{R})$ is orthogonal if $O^t O = I$. Equivalently, this means that the columns of $O$ are orthonormal, i.e., that they are orthogonal and have length $1$.

However, note that only one matrix appears in this second definition. We just say that $A$ is orthogonal or not, not that $A$ is orthogonal to $B$. So this is not the definition used by the author of your book.

• Thank you for your answer. But, given these definitions, shouldn't it be (X1)'(X2)=I in the original example? X1 and X2 are matrices, not vectors (If I got it right.). At least if it is really about orthogonality of the matrices. Could it be that the differnce between (X1)'(X2)=0 and (X1)'(X2)=I that if "...=I" then X1 and X2 are the equal matrices while "...=0" can only be if they are not equal? – Fabian Aug 19 '14 at 18:30
• No, I think the author just wants to say that the columns of $X_1$ and the columns of $X_2$ are orthogonal as vectors in $\mathbb{R}^n$, as in the first part of my answer. – André 3000 Aug 19 '14 at 18:50