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Math Help - Generate a random matrix given a condition number

  1. #1
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    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...
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  2. #2
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    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.
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  3. #3
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    Thanks for the reply .. but condition number of D does not seem to be equal to condition number of . I took invertible matrix:
    A= [1 1; -1 2] and performed with D=.but i get the condition number as 1.5328 instead of 1.500. I cant make out why??
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