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 QTAQ 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 QTAQ will also always generate an identity matrix, which is not random at all.
Answer
Let A be an upper triangular matrix with 1s along the diagonal.
If you have a complete set of eigenvectors, then A must be the identity matrix. But an upper triangular matrix with 1s along the diagonal doesn't have a complete set of eigenvectors.
The eigenvalues are all 1, and that will still be true for QTAQ because
det(A−λI)=det(QT(A−λI)Q)=det(QTAQ−λI)
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