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Thread Subject:
Condition number of a matrix is 1 but determinant is almost 0

Subject: Condition number of a matrix is 1 but determinant is almost 0

From: Hermano Cappa

Date: 11 Aug, 2011 16:27:13

Message: 1 of 2

Hello,

I want to calculate the conditioning of a matrix, therefore I use the cond() commando in Matlab. A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned.

If I calculate the condition number of the matrix:

   -0.0053 -0.0031
    0.0031 -0.0053

the condition number is equal to 1. This means that the matrix is well-conditioned. However if I calculate the determinant of this matrix with the command det() the result is 3.7700e-005. Thus almost zero what means that the matrix is almost singular, thus ill-conditioned. Why is the condition number 1 while the matrix is ill-conditioned?

Steven

Subject: Condition number of a matrix is 1 but determinant is almost 0

From: Steven_Lord

Date: 11 Aug, 2011 16:51:40

Message: 2 of 2



"Hermano Cappa" <cappasteven@gmail.com> wrote in message
news:j20vt1$3h9$1@newscl01ah.mathworks.com...
> Hello,
>
> I want to calculate the conditioning of a matrix, therefore I use the
> cond() commando in Matlab. A problem with a low condition number is said
> to be well-conditioned, while a problem with a high condition number is
> said to be ill-conditioned.
> If I calculate the condition number of the matrix:
>
> -0.0053 -0.0031
> 0.0031 -0.0053
>
> the condition number is equal to 1. This means that the matrix is
> well-conditioned. However if I calculate the determinant of this matrix
> with the command det() the result is 3.7700e-005. Thus almost zero what
> means that the matrix is almost singular, thus ill-conditioned. Why is the
> condition number 1 while the matrix is ill-conditioned?

Let A = (1e-4)*eye(100);
What is the determinant of A?
What is the condition number of A?
Do these results surprise you?
Would you consider A to be well-conditioned or ill-conditioned?

See section 2.9 of the "Linear Equations" chapter of Cleve's "Numerical
Computing with MATLAB":

http://www.mathworks.com/moler/chapters.html

in particular page 17.

--
Steve Lord
slord@mathworks.com
To contact Technical Support use the Contact Us link on
http://www.mathworks.com

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