Perpframes, Aligners and Hangers
Linear Systems, Pseudo-Inverse
Matrix Norm, Rank One
Linear Systems &
Answer: The first thing you know is that no matter what x you use, A x is always in the column space of A, Col[A].
So if ,
the equation won't have any solution.
If you should solve where is the vector in Col[A] closest to y.
The solution to
is called the least squares solution
to A x = y.
Here's what you do to solve the system:
Step 1: Find the vector in Col[A] closest to y.
is an orthonormal basis for Col[A], it's easy to compute .
Step 2: Let .
If the system has a solution, then you've just found it.
If the system has no solution, then you've done the best you can -- you've found the least squares solution.
Here's a check on your solution:
If you set , then
Note that you can compute using the matrix multiplication
is called the pseudo-inverse of A.
The least squares solution to A x=y is .
Other least squares solutions have the form
Since the SVD gives you an orthonormal basis for N[A], you know any least squares solutions to A x=y can be expressed as
If is an SVD of A,
is the reduced SVD for A.
You get the reduced SVD from the full SVD by keeping only
You can tell that the reduced SVD equals the full SVD (and so still equals A), since the two decomposition agree on the basis .
The least squares solution to A x = y is given by
As it turns out the absolute value of the determinant of A,
equals the product of the singular values.
Thus, if ,
then you know that all of the singular values are positive.
This tells you that A has rank n and so is invertible.
If all the singular values are positive, then you can compute
(1) The inverse of a product is the product of the inverses in reverse order.
(2) The aligner matrix
takes the perpframe
to the standard basis.
A similar argument explains why .
To undo the stretcher matrix
stretches the standard basis vectors by .
(3) The definition of .
Comment: You've just shown that if A is invertible, then .
In terms of the and ,
a. express the reduced SVD for A as a product of three matrices
b. express the pseudo-inverse, as a product of three matrices
c. express a solution to
d. express all solutions to
2. Suppose is a full SVD for A.
a. What are the possible values for the determinant of A?
b. In terms of
as the product of three matrices.