Is There a Name for the Decomposition of a Partitioned Orthogonal Matrix?

In summary, the theorem states that if V_1 is a matrix with orthonormal columns, then it can be extended to a larger orthogonal matrix V by adding a new set of columns V_2. This is known as a "partitioned orthogonal matrix" and is a standard result from introductory linear algebra. The proof involves extending the orthonormal columns of V_1 to a basis of \matbb{R}^n and then using the Gram-Schmidt process.
  • #1
Dafe
145
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"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.
 
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  • #2


This may be easier to see if you rephrase the problem. The columns of [tex]V_1[/tex] form an orthonormal basis for an r-dimensional subspace of [tex]\matbb{R}^n[/tex], and it is a standard result from the theory of Hilbert spaces that you may extend an orthonormal basis for a subspace to the whole space. The functional analyst in me would use Zorn's lemma to show that every orthonormal set (the columns of [tex]V_1[/tex]) is contained in an orthonormal basis of the whole space, but this is overkill in the finite-dimensional case. In this case, I'd simply find a basis of [tex]\mathbb{R}^n[/tex] that contains the columns of [tex]V_1[/tex] (using your favorite argument) and then use the Gram-Schmidt process. Not that if you let the columns of [tex]V_1[/tex] be the first r vectors in the Gram-Schmidt process, they will remain unchanged (because they are already orthogonal).
 

What is a partitioned orthogonal matrix?

A partitioned orthogonal matrix is a type of matrix that is composed of smaller submatrices that are also orthogonal matrices. It is used to represent linear transformations and can be useful in solving systems of linear equations.

How is a partitioned orthogonal matrix different from a regular orthogonal matrix?

A partitioned orthogonal matrix is made up of smaller submatrices, while a regular orthogonal matrix is a single matrix. Additionally, the submatrices in a partitioned orthogonal matrix may not be square, whereas all submatrices in a regular orthogonal matrix are square.

What is the purpose of using a partitioned orthogonal matrix?

A partitioned orthogonal matrix can be useful in solving systems of linear equations and representing linear transformations. It can also simplify calculations and make it easier to understand the structure of a larger matrix.

How do you determine if a matrix is a partitioned orthogonal matrix?

To determine if a matrix is a partitioned orthogonal matrix, you can check if all of its submatrices are orthogonal matrices. This means that the submatrices must be square and their columns must be orthogonal to each other.

What are the properties of a partitioned orthogonal matrix?

Some properties of a partitioned orthogonal matrix include:
- All submatrices are orthogonal matrices
- Each submatrix has an inverse that is also orthogonal
- The determinant of the partitioned orthogonal matrix is equal to the product of the determinants of its submatrices

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