- #1

Dafe

- 145

- 0

**"Partitioned Orthogonal Matrix"**

Hi,

I was reading the following theorem in the Matrix Computations book by Golub and Van Loan:

If [tex]V_1 \in R^{n\times r}[/tex] has orthonormal columns, then there exists [tex]V_2 \in R^{n\times (n-r)}[/tex] such that,

[tex] V = [V_1V_2] [/tex] is orthogonal.

Note that [tex]ran(V_1)^{\bot}=ran(V_2)[/tex]

It also says that the proof is a standard result from introductory linear algebra.

So I picked up my copy of Introduction to linear algebra by Strang and did not find this.

I then looked in the Matrix Analysis book by Carl D. Meyer, and here he mentiones this under the name "partitioned orthogonal matrix". I did not find a proof though.

Is there a proper name for this "decomposition"?

Thanks.