How to construct a 3*3 matrix with good quality

[Crammer007]

Dear all,

 

Would you please give me some guidance on the following simple question? Thanks!

 

Question description: I have a subset of vectors (1 by 3). 3 vectors will be drawn from it to construct a matrix (3*3). Of course, different matrixes will be produced if different 3 vectors are used. Some matrixes are close to singular while some are far away from singularity.

 

My question: What rule can be used to choose 3 vectors which make the matrix far away from singularity? If the subset has 9 vectors (1*3), are there 3 best vectors which can construct a best matrix? How to find the 3 best ones?

[Dave]

You need to give a quantitative description of what you mean by 'best'; what precisely are you looking for?

[John Cuthber]

I should knoew better than to try to answer this 'cause I'm not a mathematician but here goes.

I guess by "best" you mean with a determinant that is as far from zero as possible.

The determinant of a matrix coresponds to the volume of the object produced by that matrix operating on the unit cube, and that will be maximised (I think) if the vectors are as big as possible and as near normal to one another as possible. Certainly , if 2 of them are colineear you will have problems.

 

Of course, with a decent computer, checking all the combinations wouldn't take long.

[D H]

If your goal is to construct the "best" set of unit vectors (i.e., an orthogonal matrix), you only need to consider pairs of vectors because a basis for [math]\mathbb R^3[/math] can be constructed from any two non-collinear 3-vectors. This simplifies the problem significantly: Simply choose the pair of vectors [math](\mathbf v_i, \mathbf v_j)[/math] that maximizes [math]\frac{|\mathbf v_i \cdot \mathbf v_j|}{||\mathbf v_i||\,||\mathbf v_j||}[/math]

[Crammer007]

If your goal is to construct the "best" set of unit vectors (i.e., an orthogonal matrix), you only need to consider pairs of vectors because a basis for [math]\mathbb R^3[/math] can be constructed from any two non-collinear 3-vectors. This simplifies the problem significantly: Simply choose the pair of vectors [math](\mathbf v_i, \mathbf v_j)[/math] that maximizes [math]\frac{|\mathbf v_i \cdot \mathbf v_j|}{||\mathbf v_i||\,||\mathbf v_j||}[/math]

 

Dear D H,

 

Thanks a lot for your guidance:) I got your idea but I think the 'maximize' should be replaced by 'as close to zero as possible'. For example, given a pair of orthogonal vectors (0,1) and (1,0), their dot product should be 0*1+1*0=0. Am I right?

 

Thanks a lot for your great help:)

 

Crammer007

[D H]

Right. I was thinking of cross product but wrote dot product.

[Crammer007]

Dear John Cuther,

Thanks a lot for your great idea:) As Dave required, you give a quantitative description. I think it is easy to be applied. Based on your idea, I have a simple example below. Given two vectors (1,0) and (1, 0.0001). Obviously, they are almost parallel vectors and their determinant is 0.0001. If (1,0) is replaced by (10000,0), the determinant is 1. Thus the matrix constructed by the latter vectors is much better than the former. Am I right?

 

Thanks a lot for your guidance:)

Crammer007

 

Dear Dave,

 

Thanks a lot for your great suggestion:) What you suggested is that I want to know. Namely, which quantities can be used for the deciding rule? Please read the reply from John Cuther and D H. They gave great ideas:)

 

Thanks a lot!

Crammer007

Edited July 21, 2008 by Crammer007
multiple post merged

[John Cuthber]

Glad to have been able to offer some help. Also glad to have finally found some sort of use for the stuff that I learned about determinants about 20 years ago.