Find Intersection of 3x2 Matrices Using QR Factorization

• blabbate
In summary, the conversation discusses using the full QR factorization of two matrices to build a new matrix called Q', which contains the normals to the planes described by the original matrices. By doing a full QR of Q', the third column of Q is found to be orthogonal to both of the normal vectors, making it the intersection of the two planes. This is confirmed by a statement in Chapter 7 of the book, and the process of finding this vector is similar to finding the normal vector. The participants also discuss their thoughts and ask for feedback.
blabbate
Figured it out.

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By any chance did you do following:

Do the full QR factorization of the two matrices. Using the third column from each 'Q' matrix, build a new matrix, call it Q'. It is composed of the normals to the planes described by the original two matrices. Do a full QR of Q'. The new QR has a third column of Q that is orthogonal to both of those normal vectors. So it is therefore in the intersection.

Is this correct? or did I make a bad assumption?

It seems that we are all in CSE 6643, which also seems all good according to the syllabus and working together. So, qiaoshiya, this seems like a very reasonable assumption. From your thoughts, and talking with Prof Alben, I went back through Chapter 7. On page 50 the text states "Notice that in the full QR factorization, the columns of q_j for j>n are orthogonal to range(A)." That means that the third column of Q should basically be equivalent to cross(x1, y1), which is one way of identifying a plane (use the plane's normal vector). With this, then using the 2 third columns of the Qs, then the third QR factorization would result in a vector that is perpendicular to both of the first two plane identifying vectors. That is exactly what we're are looking for.

When I first saw this problem I went through (mostly) the exercise of finding the the final vector, since I knew how to do that. Now reading that line from page 50, the two processes seem to be identical.

Other thoughts from anyone?

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1. How is QR factorization used to find the intersection of 3x2 matrices?

QR factorization is a method used to decompose a matrix into an orthogonal matrix (Q) and an upper triangular matrix (R). By multiplying the inverse of Q with the 3x2 matrices, we can find the intersection point between the two matrices.

2. What are the benefits of using QR factorization over other methods?

QR factorization is a stable and efficient method for solving systems of linear equations. It also allows for easy computation of the inverse and determinant of a matrix, making it a useful tool in various scientific and engineering applications.

3. Is QR factorization limited to 3x2 matrices?

No, QR factorization can be used for matrices of any size. However, for the purpose of finding the intersection of two matrices, the matrices must have the same number of columns.

4. Can QR factorization be used for non-square matrices?

Yes, QR factorization can be used for non-square matrices. However, the QR decomposition will result in a rectangular matrix instead of a square matrix.

5. Are there any limitations to using QR factorization for finding the intersection of matrices?

One limitation of using QR factorization for finding the intersection of matrices is that it can only be used for linearly independent matrices. If the matrices are linearly dependent, the QR decomposition will fail and another method must be used.

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