Mjollnir

Over Specified

Sampled data often provides more equations than there are unknowns. Let's look now at the over-specified problem. We'll use the same original equations but drop out the z values.

12123
122100
2-10010
311001
110
201

We'll apply the same row and column operations to get at the "inverse". Begin with subtracting one of column 1 from column 2.

12123
120100
2-11010
310001
11-1
201

Add row 3 to row 2 and subtract 2 of row 3 from row 1.

12123
10010-2
201011
310001
11-1
201

Now swap rows 1 and 3.

12123
110001
201011
30010-2
11-1
201

The third row is identically 0, as expected, so we'll multiply the lower left matrix and the top two rows of the upper right.

12123123
11-1001=0-10
201011011

This is now our A-1. We'll confirm this by multiplying it times the A matrix.

1231212
10-102210
2011-10=01
311

With the A-1 • A being the identity matrix, there's not much point in trying to find a null-space. There's not much point in calculating the (A-1 • A) • A-1 or the A • (A-1 • A) because anything times identity or identity times anything is itself. However, it is worth noting that A • A-1 is not identity.

12123123
1220-10002
2-10011=010
311001

I may come back to this if I can find the time.

For the next part we'll do the same using row and column operations on the under-specified problem.