What is the general algorithm for computing the null space of a 3x3 matrix?

In summary, the conversation discusses two issues related to solving linear equation systems. The first issue is about a general algorithm for computing the null space using gauss-jordan elimination, and the second issue is about a closed form solution for finding the eigenvector of a 3x3 matrix. While there are known algorithms for both problems, there may be more efficient methods that require further research.
  • #1
junglebeast
515
2
There are 2 issues I want to talk about in this post.
(1) General algorithm for gauss-jordan elimination computation of null space
(2) Closed form solution to 3x3 null space

Following the example here,

https://en.wikipedia.org/wiki/Kernel_(linear_algebra)
I thought a general algorithm to compute the null space would be to

1) augment with 0 vector on the right
2) compute gauss-jordan elimination
3) take 2nd to last column, and fill in extra elements with 1's to get the null space

This works in the example provided there. However, on the next example, step 3 needs to be changed...

1, 0, 1
2, 1, 3
1, 1, 2

which has a null space of 1, 1, -1

using gauss-jordan elimination, the closest I can get is

1, 0.5, 1.5, 0
0, 1, 1, 0
0, 0, 0, 0

x1 = -0.5 x2 - 1.5 x3
x2 = -x3
[1, 1, -1]

This gives me the right null space, but step #3 of my above method clearly wasn't right..how can I generalize step 3 into a straight-forward algorithm?

Now onto my second issue.

I found a method to compute the eigenvector corresponding to an eigenvalue of a 3x3 matrix closed form. It is simply:

a1*a5 - a2*(a4-e)
a1*a2 - a5*(a0-e)
(a0-e)*(a4-e) - a1*a1

Removing the 'e's, this is essentially a short cut to get the null space. However it doesn't seem to work for non-symmetric matrices. I feel like there should be a similar method that works for non-symmetric 3x3's...which could be used to avoid the SVD method in this case
 
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  • #2
Algorithms to solve linear equation systems are well known and are part of basic computer science courses. The fact that there might be examples with faster algorithms is well known, too, and we cannot reason upon examples. This doesn't allow a generalization which would be needed to talk about an algorithm.

If you are interested in the subject, then you might want to read about the improvements on the matrix exponent: https://en.wikipedia.org/wiki/Strassen_algorithm#Asymptotic_complexity
 

FAQ: What is the general algorithm for computing the null space of a 3x3 matrix?

What is the null space of a 3x3 matrix?

The null space of a 3x3 matrix is the set of all vectors that when multiplied by the matrix result in a zero vector. In other words, it is the set of all solutions to the equation Ax=0, where A is a 3x3 matrix and x is a vector.

How is the null space related to the columns of a 3x3 matrix?

The null space of a 3x3 matrix is the orthogonal complement of the column space of the matrix. This means that the null space contains all vectors that are perpendicular to the columns of the matrix.

What is the dimension of the null space of a 3x3 matrix?

The dimension of the null space of a 3x3 matrix is the number of free variables in the system of equations represented by the matrix. This can be determined by finding the rank of the matrix and subtracting it from the number of columns in the matrix.

How do you find the basis for the null space of a 3x3 matrix?

To find the basis for the null space of a 3x3 matrix, you can use the reduced row echelon form of the matrix to identify the free variables. Then, you can set each free variable to 1 and the rest to 0 to find a set of linearly independent vectors that span the null space.

What is the significance of the null space in linear algebra?

The null space plays a crucial role in understanding the solutions to systems of linear equations. It represents all the possible solutions to Ax=0, which is a homogeneous system of equations. The dimension of the null space also provides important information about the rank and invertibility of the matrix.

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