Matrix Action 
Perpframes, Aligners and Hangers 
Matrix Subspaces  
Linear Systems, Pseudo-Inverse 
Condition Number 
Matrix Norm, Rank One 
Data Compression 
Noise Filtering 
Todd Will
UW-La Crosse
 Barely Full Rank 
Robust Rank

Round off error, 

Condition Number

Max Stretch 

Bill and Ted's misadventure.

Bill and Ted are chemists who wish to measure the number of grams of each of three different compounds A,B,C held in a single solution. 


  • a be the number of grams of A, 
  • b be the number of grams of B, 
  • c be the number of grams of C. 
By performing complicated experiments Bill and Ted are able to measure four relationships between a,b,c,d which they record in the matrix below: 


"Shoot!", says Bill. 

"We've got four equations and only three unknowns. 

Everyone knows you only need three equations to solve for three unknowns. 

Let's just delete the last equation." 

Ted agreed with Bill and so they decided to solve the system: 


"Hold on", says Ted. "We can only solve this equation if the coefficient matrix is invertible." 

After a lot of work Ted computes 


Seeing that the determinant is non-zero, Bill and Ted feel confident in computing 


Bill and Ted were pleased with their results, but they realized that the numbers they recorded on the right hand side of their equations, [Graphics:conditiongr21.gif],  were only accurate to the nearest tenth. 

They decided to round their numbers and try the calculation again: 


"Noooooooooo!", screamed Bill, "how could this happen?" 

"We've changed our matrix equation only a little but now we get a wildly different answer." 

Explain to Bill and Ted what went wrong.

Start by finding an SVD for the coefficient matrix [Graphics:conditiongr6.gif]

An SVD for A is [Graphics:conditiongr7.gif]

Since A has three non-zero stretch factors, the rank of A is three and A is invertible. 

But that smallest singular value is so small that A is nearly a rank two matrix. 

This spells trouble. 

Using the SVD method you can compute 


Now let y be the original right hand side and let y' be the rounded right hand side. 

You need to explain why [Graphics:conditiongr9.gif] is so big. 

Well, [Graphics:conditiongr10.gif]

Now the matrix [Graphics:conditiongr11.gif] stretches vectors parallel to [Graphics:conditiongr12.gif] by a factor of 5000. 

So if [Graphics:conditiongr13.gif] is parallel to [Graphics:conditiongr14.gif], then [Graphics:conditiongr15.gif]

This says that changes in the right hand side can get magnified up to 5000 times when you solve for x

Explain to Bill and Ted how to fix their problem.

First, explain to Bill that only mathematicians are silly enough to think that three equations are good enough to solve for three unknowns. 

In the real world you want all the data you can get. 

Here's the original system with the fourth equation returned. 


Now a reduced SVD for the coefficient matrix is 



This matrix is a fully robust rank 3 matrix. 

Since the singular values of A are each roughly 100, the singular values of the pseudo-inverse, [Graphics:conditiongr19.gif], will each be roughly [Graphics:conditiongr20.gif]

This tells you that if [Graphics:conditiongr21.gif] is the original right hand side 
and [Graphics:conditiongr22.gif] is the rounded right hand side, 
then [Graphics:conditiongr23.gif]

So the theory tells you that rounding y to y' won't affect the solution to the system. 

That's the theory, now here are the facts: 

Value for a,b,c using the original right hand side. 


Value for a,b,c using the rounded right hand side. 


The theory holds up! 

Theorem: If A has singular values [Graphics:conditiongr26.gif]
then [Graphics:conditiongr27.gif].

Proof: Use the orthonormal basis [Graphics:conditiongr28.gif] to write [Graphics:conditiongr29.gif]

Then the fundamental equations tell you [Graphics:conditiongr30.gif]

So [Graphics:conditiongr31.gif][Graphics:conditiongr32.gif] 



(1) If [Graphics:conditiongr35.gif] is an orthonormal basis, then 

Comment: You can use this theorem to give an upper bound on the difference between the solution to [Graphics:conditiongr37.gif] and [Graphics:conditiongr38.gif] 


1. Suppose that Bill and Ted rounded the right hand side of their equation to the nearest integer 

and then solved for [Graphics:conditiongr40.gif]

Give an upper bound on [Graphics:conditiongr41.gif]

Hint: [Graphics:conditiongr42.gif][Graphics:conditiongr43.gif] [Graphics:conditiongr44.gif] 

2. If A has singular values [Graphics:conditiongr45.gif], then the condition number of A is [Graphics:conditiongr46.gif]

Compute the condition number for 

  • the original coefficient matrix........ 

  • the coefficient matrix with the last row deleted....... 

A matrix with a large condition number is called ill-conditioned and often displays extreme sensitivity to rounded like Ted and Bill saw.

© 1999 Todd Will
Last Modified: 03-Mar-1999
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