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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,

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

(1) General algorithm for gauss-jordan elimination computation of null space

(2) Closed form solution to 3x3 null space

Following the example here,

### Kernel (linear algebra) - Wikipedia

en.wikipedia.org

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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