B.1)   SVD Matrix analysis: Writing a 2D matrix as the product of a  hanger, a stretcher and an aligner

SVD analysis tells you that every matrix can be duplicated with matrix maker ingredients: aligner, stretcher and hanger. If you want to find out how to do this for any 2D matrix, jump in right now.

B.1.a.i)

Here's  a random 2D matrix

Duplicate this matrix by coming up by with an aligner frame, stretch factors and a hanger frame so that
     A = hanger.stretcher.aligner

Answer:

First go with a cleared aligner frame:

Now go after an that makes
           :

Set s to be the result:

Plug this s into the cleared alignerframe to get your aligner:

The stretch factors are:
           
and
            
so your stretcher is:

Your hangerframe is:
         
         

This makes it automatic that
A.alignerframe[1] = xstretch hangerframe[1]
and
A.alignerframe[2] = ystretch hangerframe[2]

So your hanger is:

Try them out by comparing A with:

Success!
See it happen by looking at the unit circle and the alignerframe:

Just as it should be.

B.1.a.ii)

What is SVD analysis of a 2D matrix A?

Answer:

You just did it.
SVD analysis of a given 2D matrix A is the act of duplicating A with matrix maker ingredients
         .

B.1.a.iii)

Why does SVD analysis work?

Answer:

Given a 2D matrix A, the hanger frame  
            
The aligner frame  
            
and stretch factors
              and
satisfy
          
    .
This tells you that
     
because
            
is a perpendicular frame.
Once you realize this, you go with a cleared aligner frame
    
and you come up with an s that makes
           .
and you use this s to define your aligner frame.         
  
From this point on everything takes care of itself:
    
Your stretch factors are:
           
and
            

Your hangerframe is:
         
         

This makes it automatic that
A.alignerframe[1] = xstretch hangerframe[1]
and
A.alignerframe[2] = ystretch hangerframe[2]

B.1.a.iv)

The discussion above hinges on going with
           
and coming up with an s that makes
           .
How do you know this is possible?

Answer:

Go with a cleared matrix A:

Go with a cleared aligner frame:

The job is to explain why at least one solution s of
           
is gauranteed.
To do this, look at:

This is the same as:

And this is the same as:

This comes from applications of the identities

.

This signals that saying that
           
is the same as saying that
             
So saying that
           
is the same as saying that
           
And this the same as saying that
           
And this the same as saying that
           
And because takes on all values from -infinity to + infinity, s is guaranteed.

B.1.a.v)

Can you get more than one aligner frame for a given matrix?

Answer:

The answer depends on how anal you want to be.

Now go after the 's that make
           :

See all four perpendicular frames:

Four right hand aligner frames - but they are just different versions of the first one.
Any of these will work very satisfactorally. And if you want to go with left hand aligner frames you can go with these:

Now you've got eight aligner frames..
But one is enough.
That's why most folks are perfectly pleased to go with the first one: