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I want to generate a random matrix whose all eigenvalues are equal to one. How can I do it?
I know one method to generate matrices with given eigenvalues is to generate a random orthogonal matrix $Q$ and then construct some diagonal matrix $A$ with desired eigenvalues on the diagonal, then $Q^T A Q$ will be a random matrix with desired eigenvalues. However if I want all eigenvalues equal to 1, then $A$ must be an identity matrix, and then $Q^T A Q$ will also always generate an identity matrix, which is not random at all.

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  • $\begingroup$ What conditions do you have on the distribution? Must all such matrices be contained in the support of the distribution, for example? $\endgroup$ – Travis Jul 10 '15 at 10:18
  • $\begingroup$ For $A=(a_{ij})_{i,j=1}^n$ choose $a_{ii}=1$ for all $i$, and $a_{i+1,i} = 0\text{ or }1$ randomly. $\endgroup$ – Mher Jul 10 '15 at 10:24
  • $\begingroup$ "one method to generate matrices with given eigenvalues ...(etc)" applies only to diagonalizable matrices (even more restrictive: normal matrices). But if a diagonalizable matrix has all ones eingevalues, then it's the identity. Then that method is useless here. $\endgroup$ – leonbloy Jul 10 '15 at 21:28
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Let $A$ be an upper triangular matrix with $1$s along the diagonal.
If you have a complete set of eigenvectors, then $A$ must be the identity matrix. But an upper triangular matrix with $1$s along the diagonal doesn't have a complete set of eigenvectors.
The eigenvalues are all $1$, and that will still be true for $Q^TAQ$ because $$\det(A-\lambda I)=\det(Q^T(A-\lambda I)Q)=\det(Q^TAQ-\lambda I)$$

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The orthogonal matrices have nothing to do here.

Let $N$ be the set of nilpotent $n\times n$ matrices. It is not difficult to show that a generic matrix in $N$ (for instance, a randomly chosen nilpotent matrix is almost surely ...) is similar to $J_n$, the nilpotent Jordan matrix of dimension $n$. Thus we can proceed as follows:

Choose randomly $P=[p_{i,j}]\in M_n(\mathbb{R})$ (for instance the $p_{i,j}$ follow the law $\mathcal{N}(0,1)$ and are independent) ; $P$ is almost surely invertible. Finally, an answer to your question is $P(I_n+J_n)P^{-1}$.

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  • $\begingroup$ The reason for orthogonal matrices is computational rather than mathematical. It is much easier to compute transpose than inverse. $\endgroup$ – Sunny88 Jul 11 '15 at 10:57
  • $\begingroup$ @ Sunny 88 , I understand; yet, if you consider $B=Q^TAQ$, then the "angles" between the eigenvectors of $B$ are the same as the "angles" between the eigenvectors of $A$; thus you must also randomize $A$. $\endgroup$ – loup blanc Jul 11 '15 at 12:09

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