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I generate a random matrix. It has this general form:

$$\mathbf{B}=\left[ \matrix{ \mathbf{A}_1&\mathbf{A}_2&\ldots&\mathbf{A}_{p-1}&\mathbf{A}_p \\ \mathbf{I}_M&\mathbf{I}_M&\ldots&\mathbf{I}_M&\mathbf{O}_M \\ \vdots &\vdots &\ddots& \vdots& \vdots\\ \mathbf{I}_M&\mathbf{I}_M&\ldots&\mathbf{I}_M&\mathbf{O}_M \\ } \right]:(pM \times pM)$$

in which $\mathbf{A}_i$ is $M \times M$ for $i=1,\ldots,p$ and $\mathbf{O}_M$ ($M\times M$) and $\mathbf{I}_M$ ($M\times M$) are zero and identity matrices.

The elements of $\mathbf{A}_i$ are random (they are from a multivariate normal distribution). The eigenvalues of $\mathbf{B}$ should be less than one and I don't want to repeat random number generation process until this happens. I want to change some elements of a generated matrix (whose at least one of its eigenvalues is larger than one) so that all eigenvalues become less than one. Is there any way to do so?

(I think I should answer this question: If I change the $B(i,j)$, how will eigenvalues of $B$ be affected? But I don't know the answer).

edit: I have this idea: What if I generate a set of random eigenvalues and then generate my matrix based on them. I don't know how to proceed.

Thanks.

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    I assume you mean eigenvalues in the range $[0,1]$, otherwise $-B$ would do in case all eigenvalues of $B$ are positive...2012-11-28
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    in fact the eigenvalues in the the unit circle, since there is the possibility of complex numbers.2012-11-28
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    The pattern isn't quite clear -- are all the lower matrices $\mathbf I_M$ except $\mathbf O_M$ in the last column? Also, I think you need to say something about the type of modification you'd like -- in what sense should the resulting matrix be close to the original randomly generated $\mathbf B$? One option would be to just scale $\mathbf B$ down.2012-11-28
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    @joriki: Thanks. I edited the pattern. You are right; If I define $\mathbf{C=B}/10$, then its eigenvalues will be $1/10$-th of $\mathbf{B}$. But is there a way to change the minimum number of elements?2012-11-28
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    @Ron: I'd be surprised if it's tractable to find the minimal number of elements to change.2012-11-28
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    @Ron: I'm confused. Since the eigenvalues of $B$ may be complex, do you mean you want to make the *moduli* of the eigenvalues smaller than 1, **meanwhile** the modified entries still follow a normal distribution?2012-11-28
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    @user1551; yes. Is there a way to change some of its elements in such a way that its distribution is still multivariate normal with previous parameters, but the eigenvalues ($a+bi$) are in the unit circle ($a^2+b^2<1$)?2012-11-28
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    I've been beating my head against the wall trying to do a similar thing. My application is ensuring Bayesian VAR results are stationary in a MCMC, which I think is a slightly different set up.2013-01-28

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