The Under-specified problem
On the previous page we started with two equations and three unknowns.
What if the two equations are related?
The equations
The second equation is obviously half the first equation.
The value b1 will be twice b2.
The matrix A is:
The vector x is:
The vector b is:
Inverting the singular non-square matrix
To solve for the x vector, we augment the matrix for row and column
operations as before:
Subtract column 1 from column 2.
Subtract 2 of column 1 from column 3.
Divide 2 through row 1.
Subtract row 1 from row 2.
Now that the original matrix is reduced to Identity, we multiply the C
and R matrices to produce the A‡.
Confirm the two generalized matrix properties
To confirm this inverse meets the first two properties, we'll construct the
(A‡ A)
first:
The null space is the unused part of the C matrix:
From this matrix it is possible to directly write the arbitrariness of the solution (the null space matrix):
We can multiply the original A matrix times this null space matrix and it should be identically zero:
But this is a distraction from the stated intent... back to proving the inverse properties.
Check the product A A‡ A:
And for the second property, check A‡ A A‡
It's a trivial result with this simple set of equations, but the same applies to any set of equations.
The process of going from the null space matrix directly to the parametric or symmetric equations of lines or planes
is on this page.
In the next section we look at applying this method to the over-specified problem.