Generate a random matrix given a condition number

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January 1st 2010, 12:56 AM
mathfreak

Generate a random matrix given a condition number

If I am given the condition number , how can i generate a random matrix with that condition number. The matrix can have any number of rows and columns. please help...
January 1st 2010, 06:10 AM
HallsofIvy

The "condition number" of a matrix, A, is defined as [tex]||A||||A^{-1}|| For symmetric matrices, this is equal to $\frac{\lambda_{max}}{\lambda_{min}}$ where $\lambda_{max}$ is the largest eigenvalue and $\lambda_{min}$ is the smallest.

So the simplest thing to do to take a diagonal matrix with appropriate numbers on the diagonal. For example, if the condition number is to be 3/2, a matrix with that condition number is
$\begin{bmatrix}3 & 0 \\ 0 & 2\end{bmatrix}$ .

If you wanted 3 by 3 or 4 by 4 matrices just put numbers between those on the diagonal. To get a non-diagonal matrix (so it looks like you've done more work!) take $A^{-1}DA$ of your diagonal matrix D by some invertible matrix A.
January 1st 2010, 08:24 AM
mathfreak

Thanks for the reply .. but condition number of D does not seem to be equal to condition number of http://www.mathhelpforum.com/math-he...2d9f27b2-1.gif . I took invertible matrix:
A= [1 1; -1 2] and performed http://www.mathhelpforum.com/math-he...2d9f27b2-1.gif with D=http://www.mathhelpforum.com/math-he...f4cdc40b-1.gif.but i get the condition number as 1.5328 instead of 1.500. I cant make out why??