matrix 2-norm?

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matrix 2-norm?

Postby rolo91 » Fri Nov 16, 2012 12:44 pm UTC

I'm doing some school work, related to a matlab course, and I'm having some problems regarding the 2-norm of a matrix.

I don't understand it quite well, because my results won't fit what's expected.

Im supossed to find the 2-condition of a Wilson matrix, which apparently is equal to

Code: Select all

|||a||| * |||a^-1|||

where |||a||| is the 2-norm of the matrix a.

Now, according to my notes the 2-norm of a matrix a is equal to the square root of its largest eigenvalue (in absolute terms). so, my code goes something like this:

Code: Select all

norm = sqrt(max(eig(A)))
invnorm= sqrt(max(abs(eig(inv(A)))))
result = norm*invnorm

however, when I use the matlab command cond to check the solution, the result is totally different ( my code gives a number around 50 and matlab gives me a number around 3000).

What am I doing wrong? Are my definitions of norm and condition number correct?

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Re: matrix 2-norm?

Postby Xenomortis » Fri Nov 16, 2012 1:03 pm UTC

I confess to not being entirely certain what it is you're doing, but I think you've got the norm wrong.
|A|2 = sqrt( max{|eigenvalues of A*A|})
i.e., the 2-norm of A is the squareroot of the largest eigenvalue of A*A

(A* being the conjugate transpose of A)

(Late Edit: I meant 2-norm, not determinant squared; the statement was correct, but the notation confusing)
Last edited by Xenomortis on Fri Nov 16, 2012 4:54 pm UTC, edited 1 time in total.

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Re: matrix 2-norm?

Postby rolo91 » Fri Nov 16, 2012 4:52 pm UTC

You are completely right!

That was my problem, I had a wrong definition of 2-norm in my notes. Using your definition, it works flawlessly.


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